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WORKS BY 

PROF. WILLIAM CAIN 


IN THE 

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Price 50 cents each 

No. 3. Practical Designing of Retaining- 
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PRACTICAL DESIGNING 


OF 


RETAINING WALLS, 


WITH APPENDICES ON 

STRESSES IN MASONRY DAMS 

BT 

Professor WILLIAM CAIN, A-: M., C. E. 

(I 

UNIVERSITY OF NORTH CAROLINA. 

MEM. AM. SOC. C. E. 

ILLUSTRATED. 


SIXTH EDITION, REVISED AND ENLARGED. 



NEW YORK: 

D. VAN NOSTRAND COMPANY, 

23 Murray and 27 Warren Streets 

1910 





















Copyright, 1888, 

By W. H. FARRINGTON. 


Copyright, 1910 

By D. van nostrand COMPANY. 


/O - SX9(:, 

€c/,A250S61 



PREFACE TO SIXTH EDITION. 


In discussing all available experiments on 
model retaining walls at the limit of stability, 
in Van Nostrand’s “Engineering Magazine,” 
February, 1882 , the author was led to very 
definite views relative to the direction of the 
earth-thrust on retaining walls. The old theory 
that assumed the direction of the earth- 
thrust as normal to the inner face of the 
wall, or as having in all cases a direction 
parallel to the top surface, has been utterly 
exploded by every experiment that has been 
performed; and it is hoped that the time has 
arrived for its permanent retirement. 

This work is divided into an Introduction, 
where this direction of the earth-thrust 
receives careful attention, and four follow¬ 
ing chapters, pertaining to reservoir-walls 
and the theory of retaining-walls, developed 
respectively by the graphical method, the 

analytical method, and finally by the experi- 

iii 



iv 


mental method, leading up, after the dis¬ 
cussion of all experiments available, to 
“ the practical designing of retaining-walls/* 

In the brief discussion of dams, the 
occasion is taken to develop certain well- 
known elementary principles that are com¬ 
mon to retaining-walls as well as dams. 
In subsequent chapters of this work a good 
deal of new matter is given for the first 
time ; notably in the analytical theory of 
the retaining-wall, and in the graphical dis¬ 
cussion of “ the limiting plane ” in Chap. II. 
The theory of the retaining-wall has been 
deduced, with the one assumption of a 
plane surface of rupture, from well-known 
mechanical laws; Coulomb’s “wedge of 
maximum thrust ” being incidentally proved 
in the course of the demonstration, but not 
assumed as a first principle. 

It is hoped that constructors will hail 
with delight the numerous experiments of 
Leygue and others, discussed in the fourth 
chapter, which lead to two semi-empirical 
methods which can be confidently used in 
practice for the design of retaining-walls. 
The practical tables given in this chapter 

0 


V 


are for the first time published in this coun¬ 
try, and they are believed to be of great 
value. 

In this sixth edition, Appendices II and III 
have been added, leading to the computation 
of the “Stresses in a Masonry Dam^’ on any 
plane not too near the base. The results, 
especially when taken in connection with the 
many experiments on rubber dams made in 
England by Messrs. Wilson and Gore, are 
thought to be of the highest importance. 

The aim of the author has been to prepare 
a treatise which should serve, at the same 
time, as a text-book for our engineering 
colleges, and as a manual for the practical 
engineer; and he trusts that he may have 
succeeded passably well in carrying it out. 

Wm. Gain. 

Chapel Hill, N. G., 

December, 19 (l 9 . 








TABLE OF CONTENTS 


PAGE 

INTRODUCTIOi^.1 

CHAPTER I. 

liESEIlVOIIi-WALLS.17 


CHAPTER II. 

Theory of Retaining-Walls. — Graph¬ 


ical Method.34 

CHAPTER III. 

Theory of Retaining-Walls. — Ana¬ 
lytical Method. 7 S 

CHAPTER IV. 


Experimental Methods. Comparison 
WITH Theory. The Practical De¬ 
signing OF Retaining-Walls . . . .114 

vii 







Vlll 


APPENDIX I. 

Design for a very high Masonry Dam. 164 
APPENDIX II. 

Stresses in Masonry Dams. 173 

APPENDIX III. 

Relations between Stresses at any 
Point of a Dam. 192 




PRACTICAL DESIGNING 

OP 

EETAIITING-WALLS. 


INTRODUCTION. 

1. The retaining or revetment wall is 
generally a wall of masonry, intended to 
support the pressure of a mass of earth or 
other material possessing some frictional 
stability. In certain cases, however, as in 
dock-walls, the backing or filling — as the 
material behind the wall is called—is liable 
to become in part or wholly saturated with 
water, so that the subject of water-pressure 
has to be considered to complete the inves¬ 
tigation. In cases where the filling is de¬ 
posited behind the wall after it is built, the 
full pressure due to the pulverulent fresh 
earth or other backing is experienced; and 
the wall is designed to meet such pressure, 
with a certain factor of safety, as near as \t 



2 


can be ascertained. In time the earth 
becomes more or less consolidated by the 
settling due to gravity, vibrations, and rains, 
from the compressibility of the material, 
which thus brings iuto action those cohesive 
aud chemical affinities which manufacture 
solid clays out of loosely aggregated mate¬ 
rials, and often causes the bank eventually 
even to shrink away from the wall intended 
to support it, when, of course, there will be 
no pressure exerted against the wall. 

2. Where a wall is built to support the 
face of a cutting, the pressure may be 
nothing at first, but it would be very unwise 
to make the wall much thinner than in the 
preceding case ; for it is a well-known fact 
of observation, that incessant rains often 
saturate the ground of open cuttings to 
such an extent as to brins: down masses of 
earth, whose surface of rupture is curved, 
being more or less vertical at the top and 
approaching a cycloid somewhat in section ; 
the surface of sliding being so lubricated 
by the water that the pressure exerted hori¬ 
zontally by this sliding mass is even greater 
than for dry pulverulent materials. It is, 


3 


in fact, on this account, as well as from the 
force exerted by water in freezing, and from 
the disturbing influences caused by the 
passage of heavy trains, wagons, etc., which 
set up vibrations that low^er the co-efflcieiit 
of friction of the earth, and besides add 
considerably by their weight to the thrust 
of the backing, that a factor of safety 
against overturning and sliding of the wall 
is introduced, which factor in practice gen¬ 
erally varies between two and three when 
the actual lateral pressure of the earth is 
considered. 

3. It is stated that retaining-walls in 
Canada require a greater thickness at the 
top to resist the action of frost than farther 
south where the frost does not penetrate the 
ground to so great a depth. Again, if the 
strata in a cutting dip towards the wall, 
with thin beds of clay, etc., interposed that 
may act as lubricants when wet, the press¬ 
ure against the wall may become enormous ; 
or if fresh earth-filling is deposited upon an 
inclined surface of rock, or other impervious 
material that may become slippery when the 
water penetrates and accumulates at its sur- 


4 


face, the pressure may become much greater 
than that due to dry materials. It is found, 
too, that certain clays swell when exposed 
to the air with great force; others, again, 
remain unchanged. In all such exceptional 
cases the engineer must use his best judg¬ 
ment after a careful study of the material 
he has to deal with. The theory and 
methods used in this book will not deal 
with such exceptional cases, but simply with 
dry or moist earth-filling supported by good 
masonry upon a firm foundation ; and it is 
believed the theory deduced will be of mate¬ 
rial assistance to any one who may have to 
deal with even very exceptional conditions, 
or, as in the case of military engineers, with 
the design of revetment-walls partly as a 
means of defence. 

4. When a retaining-wall fails, it is not 
generally from not having sufficient section 
for dry backing properly laid (in layers 
horizontal or inclined downwards from the 
wall), but because the earth has been dumped 
in any fashion against the wall, and no 
“weep holes” have been provided to let 
off the water that is sure in time of rains to 


5 


saturate the bank. If to this is added bad 
masonry, and a yielding foundation, or one 
liable to be washed out, the final destruc¬ 
tion of the wall can be pretty confidently 
counted on. 

5. The folio wins; little table of weisthts 
and angles of repose of various materials 
used in construction may prove of assistance, 
but in any actual case the engineer should 
determine them by actual experiment: — 



Weight per Cubic 
Foot iu Pounds. 

Angle of 
Repose. 

Water .... 

62.4 

0 

Mud..... 

102. 

0-? 

Shingle, gravel . 

90-109-120 

35°-48o 

Clay .... 

120 

140.450 

Gravel and earth, 

126 

— 

Settled earth . . 

120-137 

21°-37° 

Dry sand . . . 

90 

34° 

Damp sand . . 

120-128 

350-450 

Marl. 

100 

— 

Brick .... 

90-135 

— 

Mortar .... 

86-110 

— 

Brickwork . . 

110 

— 

Masonry . . . 

110-144 

— 

Sandstone . . . 

130-157 

- 

Granite . . . 

164-172 

— 


We may assume generally, as safe values 
for brickwork, 110 pounds per cubic foot; 









6 


and for walls, one-half ashlar and one-half 
rubble backing, of granite 142 pounds, and 
of sandstone -120 pounds per cubic foot, 
though the last two values are generally 
exceeded. For ordinary earth or sand filling 
the angle of repose can be taken at one and 
one-half base to one rise, or a slope of 
33°42' with weights per cubic foot varying 
from 100 to 130. 

It is always advisable, where practicable, 
to put a layer of shingle next the wall, and 
to consolidate the layers of the filling by 
punning or other means, so as to reduce the 
natural slope as much as possible. 

With a well-built wall, designed after 
methods to be given ; having a good foun¬ 
dation-course, larger than the body of the 
wall, to better distribute the pressure, and 
resist sliding, and backed as described ; with 
• iveeping holes near the bottom at intervals, 
— there should be no fear of failure under 
ordinary conditions. 

6. It would take us too far to enter into 
the history of the theory of the retaiiiing- 
wall. On this point see an interesting article 
by Professor A. J. DuBois in the “ Journal 


7 


of the Franklin Institute” for December, 
1879, on “ A New Theory of the Retaining- 
Wall.” In this work three methods will be 
developed and tested by the expeiiments 
recorded. Two of these metliods are 
founded on the recent extended experi¬ 
ments of Leygue (‘■‘Annales des Pouts et 
Chaussees ” for November, 1885) and 
others, and the third is deduced by aid of 
the mechanical laws of stability in a granu¬ 
lar mass. 

7. In case a wall moves forward, how¬ 
ever little, or there is settling of the earth 
behind it, the earth generally rubs against 
the back of the wall, thus developing fric¬ 
tion. There are, however, certain inclina¬ 
tions of the back of the wall that will be 
specially examined in articles 28-31, for 
which the earth sooner breaks along some 
interior plane, in its mass, than along the 
wall, so that a certain wedge of earth will 
move with the wall as it overturns or 
tends to move. For all other cases, which 
include nearly all the cases in practice, 
there will be rubbing of the earth against 
the wall, so that the earth-thrust against 


8 


the wall must be assumed to make, with 
the normal to the wall, an angle equal to the 
co-efficient of friction of earth on wall, 
unless this is greater than for earth on earth, 
in which case any slight motion of the wall 
forward will carry with it a thin layer of 
earth, so that the rubbing surfaces are those 
of earth on earth. 

8. These suppositions are found to agree 
with experiments. The old theory that 
assumed the earth-thrust as normal to the 
back of the wall, or, as in Rankine’s theory, 
always parallel to the top slope, does not so 
agree, and, in fact, often gives, for walls 
at the limit of stability, the computed thrust 
as double that actually experienced. The 
true theory, therefore, includes all the fric¬ 
tion at the back of the wall that is capable 
of being exerted. This friction, combined 
with the normal component of the thrust, 
gives the resultant earth-thrust inclined 
below the normal to the back of the wall at 
the angle of friction to this normal.^ 


^ In Annales des Fonts et Chaussees for April, 1887, 
M. Siegler has given the results of some simple experiments 
proving the existence of a vertical component of the earth* 



9 


9. Rankine’s assumption that the direc¬ 
tion of the earth-thrust is always parallel to 
the top slope applies only to the case of an 
imaginary incompressible earth, homogene¬ 
ous, made up of little grains, possessing 
the resistance to sliding over each other 
called friction, but without cohesion ; of in¬ 
definite extent, the top surface being plane ; 
the earth resting on an incompressible foun¬ 
dation, or one uniformly compressible, and 


thrust against the movable side of a box filled with sand, by 
actually measuring the increased friction at the bottom of 
the movable board, held in place, caused by this vertical com¬ 
ponent. The box was one foot square at the base; and for 
successive heights of saud of one-third, two-thirds, and one 
foot, the vertical components of the thrust for earth level at 
top were 0.66 pound, 1.76 pounds, and 3.97 pounds, respec¬ 
tively. Similarly fot a box, 0.5 x 0.8 feet, filled with sand, 
but having a movable bottom supported firmly on iron blocks, 
the force necessary to move the blocks under the sides and 
under the bottom was measured; and from this the relative 
weights of sand supported by the bottom and sides of the 
box was found to be as one to one, nearly, for a height of 
sand of 0.6 foot, and about two to one for a height of 1.18 
foot, the total weights ascertained by the friction apparatus 
also checking out with the actual to within five per cent. 
Other experimenters have actually weighed the amounts held 
up by the sides and bottom, respectively. See Engineering 
News for May 15 and 29, 1886, also the issue for March 3, 
1883, on “A Study of the Movement of Sand;” also see 
article 60 following. 




10 


being subjected to no external force hut its 
oivn iceiglit. 

For such a material, the only pressure 
which any portion of a plane parallel to the 
top slope of greatest declivity can have to 
sustain is the weight of material directly 
above it; so that the pressure on the plane 
is everywhere uniform and vertical. If we 
now suppose a parallelopipedical particle, 
whose upper and lower surfaces are planes 
parallel to the top slope, and bounded on 
the other four sides by vertical planes, we 
see that the pressures on the upper and 
lower surfaces are vertical, and their differ¬ 
ence is equal, opposite to, and balanced by 
the weight of the particle. It follows that 
the pressures on the opposite vertical faces 
of the particle must balance each other 
independently, which can only happen when 
the}^ act parallel to the top surface, in which 
case only are they directly opposed. The 
pressures, therefore, on the two vertical 
faces parallel to the line of greatest declivity 
will be horizontal; and on the other two 
faces, parallel to the line of greatest de¬ 
clivity. This is Rankine’s reasoning, and 


11 


it is sound for the material and conditions 
assumed. It is likewise applicable to a 
material of the same kind, only compressible^ 
provided we suppose it deposited, as snow 
falls, everywhere to the same depth, on an 
absolutely incompressible, or a uniformly 
compressible, plane foundation, parallel to 
the ultimate top slope of the earth ; for then 
the compression is uniform throughout the 
mass, aud does not affect the reasoning. 
But if w^e suppose, as usually happens, that 
the foundation is not uniform in compressi¬ 
bility, then the earth will tend to sink where 
it is most yielding. This sinking is resisted 
to a certain extent by the friction resulting 
from the thrust of the earth surrounding the 
falling mass, so that much of its weight is- 
transmitted to the sides, as actually happens 
in the case of fresh earth deposited over 
drains, culverts, or tunnel linings which 
settle appreciably. In the case of a tunnel 
driven through old ground, most if not all 
the weight of the mass above it is trans¬ 
mitted to the sides ; at least, at first, before 
the timbering or masonry is got in. Again, 
if the mass of earth is of variable depth,. 


€ven on a firm foundation, the mass of 
greatest depth will sink most, thus trans¬ 
mitting some of its weight to the sides, so 
that throughout the entire mass the press¬ 
ure is nowhere the same at the same depth 
as assumed. The vertical pressure over a 
drain or small culvert crossing an ordinary 
road embankment is less, too, for another 
reason, where the embankment is highest. 
The earth-thrust on a vertical plane, parallel 
to the line of road, is horizontal for a sym¬ 
metrical section when the plane bisects that 
section. On combinino; this thrust with the 
weight of the material on either side, we 
see that the resultant load on the culvert is 
removed farther from the centre than if 
there was no horizontal thrust. It is on 
account of this tendency to equalize press¬ 
ure by aid of the friction resulting from the 
earth-thrust, that sand, when it can be con¬ 
fined, is one of the best foundations, whether 
in mass or in the form of sand piles. 

10. In the case of earth deposited behind 
a retaining-wall on a good foundation, the 
settling of the earth will generally be greater 
than that of the wall, so that the earth rubs 


13 


against the wall, giving generally the direc¬ 
tion of the thrust no longer inclined, even 
approximately parallel to the top slope 
(except when the latter is at the angle of 
repose), but making with the normal to the 
back of the wall an angle downwards equal 
to the angle of friction. If the wall should 
settle more than the filling, the thrust would 
at first have a tendency to be raised above 
the normal. But if such a thrust, when 
combined with the weight of the wall, passes 
outside of the centre of the base of the 
wall, the top of the wall will move over 
slightly, the earth will get a grip on the wall 
m the other direction ; so that it is plainly 
impossible for the wall (for usual batters at 
least) to overturn or slide on its base, with¬ 
out this full friction, acting downwards at 
the back of the wall, being exerted. Hence 
the theory which supposes it is safe ; for 
although it is possible that the earth may 
make the effort at times to exert the full 
thrust given by Rankine’s formula, yet this 
effort is suppressed instanter by the external 
force now introduced by the wall friction, 
which force was expressly excluded from 


14 


the Raukine theory. The exceptions to 
this rule will be noted in article 31. 

11. Weyranch’s objections to taking the 
thrust inclined at the angle (f)' of friction 
to the nornial are easily met. He says, 
Take a tunnel-arch; and if we suppose the 
pressure, as we go up from either side, to 
make always the angle 4>' with the normal, 
we shall have at the crown two differently 
directed pressures : similarly for a horizon¬ 
tal wall with level-topped earth resting on 
it. If there is no relative motion, or ten¬ 
dency to motion, the thrust in the latter 
case is of course vertical,’and in the former 
is probably vertical at the crown and in¬ 
clined elsewhere ; but if the arch or wall 
moves, and there is rubbing of the earth 
on the masonry, there is necessarily friction 
exerted ; so that the thrust at any point can 
have but one direction, making the angle (f>'' 
with the normal. 

12. Mr. Benjamin Baker, in his paper 
before the Institution of Civil Engineers, 
on the ‘ ‘ Actual Lateral Pressure of Earth¬ 
work ” (republished by Van Nostrand as 
“Science Series,” No. 56), tested an old 


15 


theory (where the earth-thrust was assumed 
to act normal to the wall) by the results 
of experiments, and found the theoretical 
pressure often double the actual. In the 
discussion which followed, not a single 
engineer so much as alluded to a truer 
theory which assumes the true direction of 
the earth-thrust, and has been known and 
used, just across the channel, since the 
time of Poncelet. 

The writer tested this theory by man}’ of 
the experiments recorded by Baker and 
some others, and found it to agree, within 
certain limits, remarkably well (see “Van 
Nostrand’s Magazine ” for February, 1882). 
These results have been carefully revised, 
and new experiments included, in the table 
given farther on, from which the reader 
can form a fair estimate of the theory as a 
working theory within certain limits that 
will be indicated. 

The reader is referred, however, to Mr. 
Baker’s essay, not only for experiences 
under ordinary conditions, but for those 
exceptional cases which seem to defy all 
mathematical analysis. In fact, the engi- 


16 


neer almost invariably has to assume che 
weights of earth and masonry, and angle 
of repose of the earth. Where there is 
water, the conditions one day may be very 
different from what they are the next, 
especially if the foundation is bad, as often 
happens ; in which case the wall will move 
over simply on account of the compres¬ 
sibility of the foundation, so that it has 
perhaps nothing like the estimated stability. 
For all such cases an allowance must be 
made over the results given for a firm 
foundation, etc., as to which no rule can 
be given. 

As water often saturates the filling, and 
perhaps gets under the wall, we must con¬ 
sider, in certain cases, water-pressure in 
connection with the thrust of the backing. 
Therefore, a short chapter on reservoir- 
walls, or dams, follows, in which many of 
the principles that must likewise apply to 
retaining-walls proper are given. 


17 


CHAPTER 1. 

RESERVOIR-WALLS. 

13. The design of reservoir-walls is a 
subject that has received the atteijtion of 
many engineers and mathematicians; but 
they are by no means agreed, except in a 
general way, upon the precise profile that 
is best to satisfy, as uniformly as possible, 
the requirements of strength and stability. 

We shall very briefly, and by the shortest 
means, point out the main principles of 
design of a dam that resists overturning 
or sliding by its weight alone, and is called 
a gravity dam^ in contradistinction to one 
built on a curve that requires the aid of 
arch action to render it stable. 

Let Fig. 1 represent a slice of the dam 
contained between two vertical parallel 
planes one foot apart, and perpendicular to 
the faces. 

When the dam is large, a roadway is 


18 


geoerally built on top, so that the faces hs 
and gi are vertical or nearly so for some 
distance down; after which the profile i& 
designed to meet certain requirements, to be 
given presently. Let us suppose that the 





^2 

■ -t-L_?-^ 

-J<- VL -^ 

TW' 




dam has been properly designed down to 
the horizontal joint d/, and that the w'eight 
of the portion above df equals TFj, regard¬ 
ing the weight of a cubic foot of masonry 
as 1, and that its resultant cuts the joint 
df at the point o. 


































19 


To design the part fabd below df by a 
rapid though tentative method, we must 
first assume the slopes db and fa corre¬ 
sponding to the depth dc; then compute the 
areas of the triangles bed and o/e, and of 
the rectangle feed. The distances of the 
centres of gravity of these areas (which 
represent volumes) from the point b are re¬ 
spectively |6c, be + ^ae, and be -f- ^ce. On 
multiplying each area by its correspond¬ 
ing arm from adding the products 
to W^{bc -f- do)^ and dividing by the sum 
of TFj (which equals the area of gkdf) 
and the portion added fabd, we find the 
horizontal distance 6m from b to where the 
resultant of the weight above joint ab cuts 
this joint. Its amount W is equal to the 
sum of the areas ( TF, + abdf), and we 
have only to combine TF acting along the 
vertical through m, with the horizontal 
thrust H of the water acting on the face 
ksdb, to find the resultant R on the joint, 
and the point n where it cuts that joint. 

There is a vertical pressure of the water 
on the part sdb; but, as it adds to the 
stability, it is generally neglected, particu- 


20 


larly as the inner face is generally nearly 
vertical. 

14. The horizontal pressure of the water 
II for the height /i, by known laws of 
mechanics, is equal to the area /i X 1 mul¬ 
tiplied by the depth of its centre of gravity 

- below the surface of the water, and by 
2 

the weight of a cubic foot of water w, 
where a cubic foot of masonry is taken as 
the unit. This pressure acts horizontally 
at ^li above the joint a6, so that its moment 
about the point n where the resultant R 

_ h h _ h^w rp. 

cuts the base ab is h . - ~ — ——• 

zoo 

moment of W about the same point is 
W X 7)111. As these two moments must be 
equal, we find the distance between the 
resultant pressures on joint ab for reservoir 
empty and reservoir full. 


mn 


h^uo 

6TF* 


The above is substantially one of the methods 
adopted by Consulting Engineer A. Fteley in the 
design of the proposed Quaker Bridge Dam. See 
bis interesting report, and that of B. S. Church, 



21 


chief engineer, with many diagrams of existing 
dams of large proportions, in “EngineeringNews” 
for 1888, Jan. 7, 14, Feb. 4, 11; also the discus¬ 
sions by the editor in the numbers for Feb. 4 and 
25, and March 3. 

15. There are three well-known condi¬ 
tions, that must hold at any joint if the 
profiles fa and db have been designed 
correctly: — 

1st, The points m and n where the re¬ 
sultants for reservoir empty or full cut the 
base ab must lie within the middle third of 
the joint or base ab. 

2d, The unit pressures of the masonry at 
the points a or b must not exceed a certain 
safe limit. 

3d, No sliding must occur at any point. 

16. The last condition is evident, and 
requires that H < Wf where / is the co¬ 
efficient of friction of masonry on masonry, 
the adhesion of the mortar being neglected. 
If is the angle of repose of masonry on 
masonry, / = tan and we must always 
have, 


— < tan ; 
W ^ 


22 


that is, the resultant M must never make 
with the normal to the joint an angle 
greater than the angle of friction. In fact, 
in practice, we should employ some factor 
of safety as 2 or 3, so that 2H or 3// should 
always be less than Wf. This third con¬ 
dition is of supreme importance at the 
foundation joints of dock-walls, which fail 
(wlien they fail at all) by sliding from 
the insufficient friction afforded by the wet 
foundation. For ordinary retaining-walls, 
too, the foundation should, when practi¬ 
cable, be inclined, so that li shall make a 
small angle with the normal to the base. 
In all cases, deep foundations are to be 
preferred, as the earth in front of the wall 
resists the tendency to slide appreciably. 

17. We shall now proceed to give a 
reason for the first condition above, and 
likewise deduce a formula to ascertain the 
unit stresses at the points a and b. 

If we decompose the resultant B at the 
point n, distant u = an from a (Fig. 1), 
into its two components H and IF, the 
former is resisted by the friction of the 
joint, and will be neglected in computing 


23 


the stresses at a and 6, though it doubtless 
affects them in some unknown manner. 
The remaining force W, acting vertically 
at w, must necessarily cause greater press¬ 
ure at the nearest edge than elsewhere on 
the joint, at least when the angle at a is not 
too acute, and the dam is a monolithic 
structure. For large dams built of stones 
in cement, it is likely that there will be 
greater pressure at the middle of the base 
than in a monolithic structure where the 
resistance to shearing or sliding along ver¬ 
tical planes is much greater than in a wall 
made up of many blocks, particularly if 
they are laid dry. But it is probably best, 
until experiment can speak more decisively 
on the point, to assume the pressure great¬ 
est at the toe nearest the resultant, and as 
given by the following theory : — 

Call I = length of joint ab 

u z=z z= distance from R to near¬ 
est toe; 

then if we suppose applied at the centre of 
the joint two vertical opposed forces, each 
equal to TF, it does not affect equilibrium. 
We can now suppose the force W acting 



24 


downwards at the centre to be the resultant 



shown by the little arrows just below joint 
ah; and that the remaining forces fF, one 
at the centre and one at ri, acting in oppo¬ 
site directions, and constituting a couple, 
whose moment is W — u)^ cause a uni¬ 
formly increasing stress, as in ordinary 
flexure (shown by the little arrows below 
the first), whose intensity at a or b is by 
known laws. 



The total stress p at the nearest toe a is 
therefore the sum of p-^ and p^^ and is com¬ 
pressive. 



• ( 1 ) 


The stress at h is of course p^ — p.^^ where 
this is not minus indicating tension, unless 
the joint can stand the tension required. 
If we call u' the distance from n to the 
farthest toe, i.e. u' = nb, we have the mo- 





25 


ment of the two weights W = W{u' — ^V), 
On substituting this value for — u) in 

the value for above, we find for the unit 
stress at 5 the identical equation (1) above, 
provided we replace u by u '; so that the 
equation is general, and applies to either 
toe, if we only substitute for u the distance 
of the resultant from that toe. The stress 
is distributed, as shown by the lower set of 
arrows in Fig. 1, where there is only com¬ 
pression on the joint as should always 
obtain. The stress is thus uniformly in¬ 
creasing from the right to the left. If the 
limit of elasticity is nowhere exceeded, it 
follows that a plane joint before strain will 
remain a plane joint after strain, as must 
undoubtedly be the rule for single rectangu¬ 
lar blocks. 

Keferring to equation (1), we see that if 
we replace u by u' = f Z, that the stress at h 
is zero, from which point it increases uni¬ 
formly to a, where its intensity, for u = 

W 

is p = 2—, or twice the mean. For greater 
1 

values of u' than fZ, the stress at b becomes 
tensile, which is not desirable; hence the 


26 


reason for condition 1 above, that the re¬ 
sultant should lie .within the middle third 
of the joint. 

If the joint cannot resist tension at all, 
and R strikes outside the middle third, the 
joint will bear compression only over a 
length and the maximum intensity at 
W 

a is now 2—. This is evident, if we treat 

Su = I' as the length of joint, and substi¬ 
tute this value for I in formula (1). There 
is now no pressure at the distance 3u = V 
from the left toe by the previous reasoning 
for the original joint and to the right of 
that point the joint will open, or tend to 
open. It is evident for full security that 
the resultant should strike within the mid¬ 
dle third some distance to allow for con- 
tinoeucies. 

18. Having computed the unit pressures 
at tlie nearest toes for reservoir full or 
empty, condition 2 requires that these 
pressures do not exceed certain limits : in 
case they do, the lower profiles have to be re¬ 
vised, and the computation above repeated, 
until all the conditions are satisfied. 


27 


In the proposed design for Quaker Bridge 
Bam, maximum pressures per square foot at the 
toes, at tile base, were limited to 30,828 lbs. at 
the back, and 33,266 lbs. at the face; these 
pressures diminishing gradually to one-half to 
within about 100 feet from the top, the total 
height of dam from the foundation being 265 
feet; the argument being that the lower parts 
could stand more pressure than the upper parts 
shortly after construction, on account of the 
cement there attaining a greater strength. Be¬ 
sides, for this unprecedented height of dam, to keep 
the lower pressures within more usual limits “it 
would be necessary to spread the lower parts in 
an impracticable manner, and to incline the slopes 
to an extent incompatible with strength.” 

It is evident that by this method of design 
there is no fixed rule by which any two computers 
could arrive at the same profile, having given the 
upper part empirically, sufficient in section to 
carry a roadway, and to resist the additional 
stresses due to the shock of waves and ice, at a 
time, too, when the mortar is not fully set. 

Such a rule is most easily introduced by 
requiring a certain factor of safety against over¬ 
turning, and, moreover, that the factor of safety 
against sliding along any plane shall not fall below 
a certain amount. It is suggested, however, that 
the factors of safety should increase from the 
foundation upwards, to make the section equally 
strong everywhere against overturning, when 


28 


allowance is made for the effects of wind and wave 
action, floating bodies, the expansive force of 
ice, or perhaps the malicious use of dynamite. 
If this is admitted, it would add one more con¬ 
dition (4) to the three previously stated, and 
Would secure greater uniformity in design. See 
Appendix. 

As to the unit pressure test (condition 2), it 
must be observed, that we know little or nothing 
as to what limit to impose; for not only is the 
stress all dead load (which would allow of higher 
unit stresses), but the unit resistance of masonry 
in great bulk is undoubtedly much greater than 
in small niasses (not to speak of tests on small 
specimens as a criterion), since the shearing off 
which follows, or is an incident to, crushing can 
hardly occur in the interior of a large mass of 
masonry. 

19. We shall find in the end, that, for 
different forms of retaiuin 2 :-walls to sustain 
earth, that a factor of safety of about 2.5 
against overturning is liiglily desirable, and 
that it will generally satisfy the middle third 
limit. In such walls this factor must be 
introduced to provide' against an actual 
increase of the earth-thrust, due to water, 
freezing, accidental loads, and above all 
to the tremors caused by passing trains or 


29 


vehicles (if these are not considered sepa¬ 
rately), which it is well-known have caused, 
by increased weight, and the increased 
pressure due to lowering the natural slope, 
a gradual leaning and destruction of walls 
of considerable stability for usual loads. 

In a very high dam this is different: 
the pressure rarely changes but little, ex¬ 
cept on the upper portions; so that, if such 
conditions were to hold indefinitely, the 
limiting unit stresses should control the 
lower profile more than a factor of safety 
against overturning. But, as pointed out 
by the editor of “ Engineering News ” (in 
the issues above referred to), a dam on 
which the fate of a city may ultimately 
depend should be designed, as far as pos¬ 
sible, to resist earthquakes also. For that 
contingency, there is a reason for the factor 
of safety against overturning and sliding 
being as great as possible throughout; and 
by putting the gravity dam in the arch 
form, convex up stream, the resistance to 
earthquake and other shocks is enormously 
increased. 

20. We have now given the general prin- 


30 


ciples that should guide in the design of 
dams, which likewise apply in the design 
of retaining-walls proper, where, however, 
the height is rarely sufficient to call for 
much, if any, change of profile, and the 
maximum pressures are usually far within 
safe limits when a proper factor of safety 
against overturning or sliding has been 
introduced, which satisfies likewise the con¬ 
dition that the resultant shall cut the base 
within the middle third. We of course 
have, as stated before, the direction of the 
earth-thrust inclined below the normal to 
the wall at the angle of friction ; otherwise, 
the methods above are applicable when the 
value of that earth-thrust has been deter- 
miued. For dock or river walls, saturated 
with water, The buoyant and lubricating 
effect of the water must he considered. 

If we suppose the filling of gravel, the 
water surrounding each stone allows free¬ 
dom of motion ; but the weight of the solid 
stones of the filling must now be taken less 
than when in air, by the weight of an equal 
volume of water, or at the rate of 62.4 
lbs. per cubic foot (or say 64 for salt 


31 


water), and the earth-thrust then found 
for the angle of repose of stone lubricated 
with water. Thus, if the weight of the 
solid stone be 150.4 lbs. per cubic foot, 
and the voids are thirty per cent, the weight 
of solid stone in water is 88 lbs. per cubic 
foot, and that of the filling 88 x .70 = 
61.6 lbs. in water, although it was 105 in 
air. 

If the wall is founded on a porous 
stratum, the weight of the masonry is sim¬ 
ilarly reduced by 62.4 lbs. per cubic foot, 
or say one-half ordinarily; but if the 
foundation is rock or good clay, ‘‘there is 
no more reason why the water should get 
under the wall than it should creep through 
any stratum of a well-constructed masonry 
or puddle-dam,” as Mr. Baker has ob¬ 
served. 

If the water cannot get in behind the 
wall, the water in front only assists the 
stability. 

It has been previously observed that 
sliding is principally to be guarded against 
in dock-walls and others similarly situated, 
which can only be done by a sufficient 


32 


weight of masonry irrespective of its shape, 
unless the foundation is inclined, which 
even in the case of piling has been effected 

Fig. 2 



by driving the piles obliquely, of course 
as nearly at right angles to the resultant 
pressure as is practicable. 

Fig. 2 represents a wall with a curved 
batter, in brickwork with radiating courses, 













33 


( 


that might be used for a quay or river-wall, 
or a sea-wall, as ships can come closer to 
the brink than in the case of a straight 
batter; besides, for sea-walls it resists the 
action of the waves better. The centre of 
gravity can be found by dividing the cross 
section up into approximate rectilinear 
figures, and proceeding as in finding the 
position of TF in Fig. 1. Its position is 
a little farther back than for a straight 
batter, which adds to its stability. But it 
is difficult to construct, the joints at the 
back are often thicker than is advisable, 
and there is probably no ultimate economy 
in its use. 


34 


CHAPTER II. 

THEORY OF RETAINING-WALLS. 

Graioliical Method. 

21. In the theory of earth-pressure that 
follows, we shall consider the earth as a 
homogeneous, compressible mass, made up 
of particles possessing the resistance to 
sliding over each other called friction, but 
without cohesion. This is a much simpler 
definition than the one that Rankine’s 
theory calls for (see Art. 9), and is more 
true to nature; the only approximation, in 
fact, consisting in neglecting cohesion, if 
we consider a homogeneous earth like dry 
sand. 

Let Fig. 3 represent a vertical section of 
a retaining-wall ABCD, backed by earth, 
whose length perpendicular to the plane of 
the paper is unity. 


35 


Assumption. We assume that the earth 
behind the wall, whether the top surface is 
a plane or not, has a tendency to slide 
along some plane surface of rupture as 
J.1, ^2, ... . 

TTigri 3 



No proof is given of this assumption, so 
that it can only be tested by experiment; 
but for the present we shall adopt it. 

In connection with the hypothesis of a 
plane surface of rupture, we shall use only 
one principle of mechanics relative to the 












3G 


stability of a granular mass, first stated 
by Rankine as follows : — 

It is necessary to the stabiliiy of a 
yranular mass, that the direction of the 
liressure between the portions into tuhich it 
is divided by any plane should not, at any 
point, male ivith the normcd to that plane 
an angle exceeding the cuigle of repose. 

This principle will alone enable us to 
ascertain the earth-thrust against any plane 
without resorting to a special principle, 
like Coulomb’s ‘‘ wedge of maximum 
thrust,” which last, however, will be in¬ 
cidentally demonstrated as a consequence 
of the above law. 

22. In Fig. 3, let us consider the. 
triangular prisms CAO, CA\, . . . , as 
regards sliding down their bases AO, 

Al, ... . 

If AF is the natural slope of the earth, 
the tendency of the prism CAF to slide 
along AF is exactly balanced by friction, 
as is well known. But if we consider 
other possible planes of rupture, lying 
above AF, as AO, ^11, . . . , we see, unless 
the wall offers a resistance, that sliding 


37 


along some one of these planes must 
occur: so that the earth exerts an active 
thrust against the wall, which must be 
resisted by it; otherwise, overturning or 
sliding would ensue. 

In case the wall is subjected to a thrust 
from left to right, as from«earth, water, 
etc., acting on BD^ and this thrust is 
sufficient to more than counterbalance the 
active thrust of the earth to the right of 
the wall, it will bring in the passive 
resistance of the earth to sliding up some 
plane as A2, and the surface of rupture 
will now resist motion upwards, in place of 
downwards as hitherto. 

In the first case, of active thrust, where 
the prism is just on the point of moving 
down the plane, we know by mechanics 
that the resultant pressure on the plane 
of rupture makes an angle 0 of friction of 
earth on earth with the normal to that 
plane and directed belotv the normal; in 
the second case, of passive thrust, the 
direction of the pressure lies above or 
nearer the horizontal than the normal, and 
makes the angle ^ with the latter. 


38 


23. In the first case, where the wall 
receives only the active thrust of the prism 
of maximum thrust, let us call G (Fig. 3) 
the weight in pounds of this prism ; S the 
resultant pressure on the surface of rupture, 
making an angle ^ with the normal to that 
plane below the normal; and E the resultant 
earth-pressure on the wall, which (except 
for cases to be noted in Art. 31) makes an. 
angle (/>' of friction of earth on wall with 
the normal to the wall below the normal, 
unless (j)' > in which case a thin layer 
of earth will go with the wall, in case of 
relative motion, and this layer rubbing 
against the remaining earth will only cause 
the friction of earth on earth, and E will 
only be directed at an angle ^ below the 
normal; supposing always that the tendency 
to relative motion corresponds. to the earth 
moving down, along the back of the wall 
AC, as in settling from its compressibility, 
or as in case of an incipient rotation of the 
wall forward, from a greater pressure on 
the outer toe or a slight unequal compres¬ 
sion of the foundation. 

It remains to find the position of the true 


39 


plane of rupture. As preliminary to this, 
we note from Fig. 3 an expeditious way of 
finding the direction of S on any trial plane 
of rupture, as Al. Thus calling w the angle 
that Al makes with the vertical AJ, the 
straight line making an angle o>) 

with any horizontal, as DC I, below that 
horizontal, is parallel to /S', since any line 
inclined at an angle w below the horizontal 
is perpendicular to Al , and S is inclined at 
an angle below that normal. In laying 
off the equal angles, it is convenient to use 
a common radius, AH^ to describe the arcs 
having A and / respectively as centres, and 
to take chord distances of the arcs (f) and co, 
and lay tliem off on the arc with 7 as a 
centre, as shown. For any other trial 
plane, as A2, we have simply to lay off the 
corresponding value of w below the angle (j> 
as before. 

24. We shall now refer to Fig. 4, to 
illustrate the general method to follow to 
find the earth-thrust E in pounds. Here 
the wall, one foot long perpendicular to the 
plane of the paper, is shown in section 
BACD^ the earth sloping at an angle from 


40 



some point on the top of the wall to the 
point marked 2, where it is horizontal. 
This is called a surcharged wall, the earth 














41 


Ijing above the horizontal plane of the top 
of the wall being called the surcharge. 

Extend the line AC of the inner face to 
0, whei-e it intersects the top slope of the 
earth; the possible prisms of rupture are 
then ^401, ^102, yl03, . . . , and we shall 
now proceed to reduce these areas to equiv¬ 
alent triangles having the same base A2. 
Draw the parallels 00', 11', 33', . . . , to 
line A’l to intersection with a perpendicular 
to A'2, passing through the point 2. Then 
the triangle ^02 is equivalent to the triangle 
-40'2, and ^12 to Al'2, so that triangle 
^O'l' is equivalent to ^01. Similarly ^126 
is equivalent to triangle A2G'A, having the 
same base, A2, and vertices in a line parallel 
to this base, giving the same altitude. Thus 
the area ^026^ is replaced by ^0'6'^ ; and 
the weight of the corresponding prism, if we 
call e the weight per cubic foot of earth, 
is ^A2 X 0'6' X e. Similarly the weight of 
^024 is ^A2 X 0'4' X e; so that if we use 
O'l', 0'2, 0'3', . . . , to represent the weights 
of the successive prisms ^01, ^02, ^03, 
. . . , on the force diagram given below, 
we have simply to multiply the value of 


♦ 




42 


given by construction, by ie,A2 to find its 
true value in pounds. 

We next lay off the successive values of 
((f> w), as in Fig. 3. Thus, with any 

convenient radius, as AO, we describe an 
arc, ogd/, and call the intersections with 
Al, A'2, . . . , ttp ttg, O 3 , . . . , respectively. 
Next, through point g on the arc in the 
vertical through A, draw vertical and hori¬ 
zontal lines, and describe an arc, hss^, . . . 
with the same radius ; then draw gs, making 
the angle below the horizontal gh (by 
making chord hs = chord fd) , and lay off 
with dividers, chords ss^, ss.,, ss^, . . . , 
equal to chords ga^, ga^, ga^, ... It is 
evident now that lines gs^, gs^, . . . , 
make the angles cf) with the normals to the 
successive planes ^ 11 , ^2,^13, . . . , and thus 
give the direction of the N’s corresponding 
to those planes. 

We now lay off with dividers on the 
vertical line gA the distances gg^, gg^, . . . 
equal respectively to OT', 0'2, 0'3', . . . , 
and draw through the points g^, g^, g^, 
parallels to the direction of E (drawm as 
before explained) to intersection with the 


« 


43 


lines ^£ 2 , , which intersections 

call Cp C 2 , Cg, . . . , respectively. 

25. It follows that the lines g.f^^ 
gTgCg, . . . , represent the thrusts E due to 
the successive prisms of rupture ^101, ^02, 
. . . , and we shall now prove that the 
greatest of these lines, which is found to 
be g^c^^ represents the actual active thrust 
upon any stable wall.^ This follows from 
the simple fact, that if we regard any 
other thrust than the maximum as the true 
one, on combining this lesser thrust, taken 
as acting to the right, with the weight of 
the wedge of rupture corresponding to the 
maximum thrust, we necessarily find that 
the resultant falls below the position first 
assumed ; so that it makes an angle with the 
normal to the corresponding plane of rupture 
greater than the angle of repose, which, by 
the principle of Art. 21, is inconsistent with 
stability. Thus, in Fig. 4, if we choose 
tc assert that any trial thrust, as g.^c^^ less 
than the maximum g^c^^ is the true one, on 


1 This method of laying off the trial thrusts, so that the 
maximum could readily be obtained, was first given by 
professor Eddy, in New Constructions in Graphical Statics. 






44 


shortening the lengths ^gCg, • • • 9 

representing superior thrusts, to the com¬ 
mon length ^ 2^21 drawing through the 
new positions of Cg, c^, . . . , straight lines 
to which thus represent the resultant 
thrusts on the planes ^3, ^4, . . . , we see 
that the new directions fall below the first 
assumed positions, and therefore make 
angles with the normals to the planes 
greater than which is absolutely incon¬ 
sistent with equilibrium. It follows that 
any thrust less than the maximum, as 
determined by the construction above, is 
impossible; and that this maximum thrust 
thus found is the actual active thrust exerted 
against the wall. In this consists what is 
known as Coulomb’s “wedge of maximum 
thrust,” which is here established by aid of 
the single mechanical principle enunciated 
in Art. 21. 

The prism of rupture in this case is 
-<4024^, the plane A4 being the surface of 
rupture. 

To find the resultant thrusts on all the 
other assumed planes, we combine the actual 
thrust found with the weight of earth lying 



45 


above the plane. Thus, extending g^c^^ 
. . . to a common length or to the 
vertical tangent to the dotted curve, the lines 
drawn from g through the corresponding 
intersections with this vertical will represent 
the thrusts on the planes Al^ ^2, . . . , 
which are thus inclined nearer the horizontal 
than the old trial values, and thus make less 
angles than 4> with the normals to their 
corresponding planes ; so that the conditions 
of stability are all satisfied, and, if the wall 
gives, sliding will only occur down the plane 
of rupture AA. , 

In the analytical method followed by 
Weyrauch, E is assumed to be constant, 
and to equal the actual thrust on the wall; 
and the real surface of rupture is taken to 
be that plane, for which the angle that S 
(Fig. 3) makes with the normal is the 
greatest (cfi) consistent with equilibrium, 
which is in agreement with what we have 
just proved 

Winkler adopts the same method, in 
preference to the Coulomb method. In 
fact, he asserts that “no author, from 
Coulomb down, has given any direct satis- 


46 


factory proof of Coulomb’s principle.” It 
is hoped that the above demonstration 
will prove complete and satisfactoiy. The 
method evidently gives the least thrust, for 
the assumed direction of that will keep 
the mass from sliding down the surface of 
rupture. 

The earth can resist a much greater 
pressure from the wall side, since a con¬ 
tinuously increasing pressure from the left 
causes all the resultants on planes ^11, ^42, 
. . . to approach the normals, then to pass 
them, and finally to lie above them, with 
the sole condition that none of them must 
make angles greater than with their 
corresponding normals (see Art. 34). 

26. To find the thrust E in pounds^ we 
multiply g^c^ to scale by ^A'l.e. Finally, 
if we know the position of E^ we combine 
it with the weight of the wall in pounds, 
acting along the vertical through its centre 
of gravity, to get the resultant on the base. 
If the upper surface of the earth is level, 
or with a uniform slope from the point 0 
(Fig. 4), then the sections of the prisms of 
rupture for various heights of the wall, or 


47 


for any values of ^40, are similar triangles, 
60 that the thrusts which vary directly 
with the weight of the corresponding prisms, 
will also vary directly as the areas of these 
triangles, or as the squares of the homologous 
lines ^0, or as the squares of the height of 
point 0 from the base AB. It follows, as 
in the case of water-pressure, that for these 
cases the resultant E of the earth-thrust 
acting along the face ^0 is found at a 
point ^^0 along ^0 from the base AB. 

For the surcharged wall it is possibly 
higher; in fact, Scheffler takes it in con¬ 
structing his tables, for all cases, at jqAO 
along ^0. But experiment indicates either 
that the thrust is overestimated for sur¬ 
charged walls, or that it acts not higher 
than at one-third the heigrht of 0 above the 
bash ; so that it will prove safe in practice 
to take the latter limit if we use the 
theoretical thrust. As to the latter, it is 
evident that cohesion (which we have 
neglected) has a greater area to act upon 
along the surface of rupture for any kind 
of surcharged wall, than for earth either 
level or sloping down from the top of the- 


48 


wall; so that we should expect the thrust to 
be somewhat overestimated when we neglect 
cohesion altogether, since the resistance to 
sliding down any plane due to it is directly 
as the area of the surface of separation. 

27. In case the earth is level with the 
top of the wall, the construction of Fig. 4 
again applies, only the line now coincides 
with the horizontal through ( 7 , and the 
reduction of areas to equivalent triangles 
is omitted, since now all the triangles have 
the same altitude, equal to the height of the 
wall. 

If, however, the earth slopes uniformly 
from the top of the wall, at a less angle 
than the angle of repose, we can assume 
any point as 2, on this slope, and effect the 
construction of Fig. 4 as before; or, better, 
we can divide this slope into a number of 
parts at 1, 2, ... , and treat 01, 02, . . . , 
successively as the bases and the perpen¬ 
dicular from A upon 02, produced as the 
common altitude; so that, using 01, 02, 
. . . , as representing tlie weights of the 
corresponding prisms on the. load line gg^ 
we have finally to multiply the value of gc^ 



49 


corresponding to the greatest thrust, by 
multiplied by this perpendicular, to get 
the maximum thrust E in pounds. 

In case the surface of the earth slopes 
indefinitely at the angle of repose^ the 
graphical method fails to find the surface 
of rupture, which analysis shows, in this 
case, to approach indefinitely to the plane 
of natural slope passing through the point 
-4, though practically it may be shown that 
planes of rupture slightly above the latter 
will give almost identically the same earth- 
. thrust, so that they can safely be used. In 
fact, it is well to state here, that, for earth- 
level at top, the surface of rupture, as 
observed in experiments with every kind of 
backing, agrees very well with theory ; but, 
as the surcharge grows higher, the actual 
surface of rupture lies nearer the vertical 
than the theoretical, and the thrust .is 
correspondingly less, particularly for walls 
leaning backwards at top, which, for a high 
surcharge, actually receive much less thrust 
than the simple theory after Coulomb’s 
hypothesis, neglecting cohesion, calls for; 
^•nd it is not surprising that it is so. But 


50 


we shall defer the comparison of numerical 
results till later. 

28. Case ivliere E does not make the angle 
<f) or (f) with the normal to the ivall. 

In Fig. 5, let AC represent the inner face 
of the wall, backed by earth sloping upwards 
from C in the direction (7—10. There 
are certain positions for the loall AC lying 
to the left of the vertical Ag^ for ivhich the 
true thrust on it is found by ascertaining 
the thrust on the vertical p)lane Ai)^ extending 
from the foot of the icall A to where it 
intersects the top slope C — 10, having 
assumed the direction of the thrust on JIO, 
after Eankine, as parcdlel to the top slope, 
and combining this thrust, acting at ^AO 
above A, loith the weight of the mass of 
earth, AOC, lying beticeen ^0 and AC, 
acting cdong the vertical through its centre 
of gravity. The thrust on AO is thus 
combined with the weight of AOC, at a 
point on AC, one-third of its length going 
from A to C. 

This direction of the thrust on ^0 par¬ 
allel to the top slope is in agreement with 
Rankine’s principle for the case of an 

















I 






I • 




if 


'*• : 

• .> 

► ^ 







V' 


4 


V.' 

K^' 

»?<( 


•f 


A 


it; 


\> » 

l- • 

i • 

1 ^ ’■ * 


V • - 

■J\ 


/• 1. 


I 


•. >1 






51 


unlimited mass of earth of the same depth 
everywhere, on an uniformly compressible 
foundation (Art. 9), and doubtless agrees 
very nearly with the direction and amount 
of the earth-thrust in ordinary cases, except 
•near comparatively rigid retaining-walls, or 
other bodies, where the direction is generally 
changed, as previously pointed out. Let 
us ascertain the limiting position of A(7, 
below which the true thrust must be ascer¬ 
tained in the manner just stated. To do 
this, we first assume the thrust on AO as 
acting parallel to the top slope, and find 
its intensity corresponding by previous 
methods ; and afterwards prove, for positions 
of AC below the limit, to be found b}^ 
construction, that no thrust on ^40 having 
a less inclination to the vertical is consistent 
with equilibrium. 

The construction necessary to find the 
thrust on AO, from the earth on the right, 
is similar to that given for Fig. 4, except 
that the top slope is now uniform, and will 
only be briefiy indicated. Thus, divide the 
top slope 0 — 10, to the right of Ag, into a 
number of parts, made equal for convenience, 


52 


and draw through the points of division lines 
from A produced on to meet the arc described 
with Ag as a radius at the points a^, . . . 

Then with ^as a centre, and gA as a radius, 
describe a semicircle as shown ; also draw 
gb horizontal, and lay off arc bs equal to 0, 
the angle of repose, and from s lay off arcs 
sSj, ss.^, . . . , equal to ga^, and draw 
the Hues gs^, gs^, • . . , from g through 
the extremities of these arcs to represent the 
directions of the resultants on the successive 
planes of rupture, which are thus inclined 
below the normals to those planes at the 
angle cf) respectively. Next, on the vertical 
gA, lay off gg^^ gg^, . . . , equal to the bases 
01, 02, of the supposed prisms of rupture 
lying to the right of and through their 
extremities draw g^V, g.^\ • • • ? parallel 
to top slope to intersection 1', 2', . . . . 
with the directions of the resultants firs' 
found. The greatest of these lines cc'’, to 
scale, represents the actual thrust ou ^10 ; 
and we have only to multiply it by 
where p is the perpendicular let fall from 
A on the top slope 0—10 produced, to scale, 
to get the pressure in pounds, if desired. 



53 


Now, if the direction of the pressure on the 
wall AC cannot be taken as usual, inclined 
below the normal to AC^ at an angle (/), 
it is (Art. 7) because, in case of motion, 
the earth does not rub against the wall 
sufficiently to develop the required friction, 
whence it must follow that the earth breaks 
along some plane as A.4, A5, . . . , to the 
left of Ag^ where the thrust is inclined at 
the angle ^ to its normal; so that this plane 
is a veritable plane of rupture^ and its 
position can be found as usual on assuming 
the direction of the thrust on AO as parallel 
to the top slope. 

In case such a plane exists between AO 
and AC^ the earth below it, if the wall 
moves, will go with the wall; further, it is 
evident that the thrust against the vertical 
plane AO, due to the wedge of rupture on 
the left, must exactly equal the thrust first 
found corresponding to the wedge of rupture 
on the right, otherwise equilibrium will be 
impossible. 

To ascertain the position of this plane of 
rupture on the left, that we shall hereafter 
call the limiting plane^ most accurately, it 


54 


is well to magnify the lines representing the 
forces as much as the limits of the drawing 
will admit of. We have consequently 
divided the top surface, OC’, into a number of 
equal parts, of which the first eight are only 
one-fourth the length of the corresponding 
parts to the right of ^40. By la^ung off the 
loads gg^^ • • • ? however, to a scale 
four times as large as just used, we have 
the lengths gg^^ gg.^^ • • • ? exactly four 
times the lengths 01, 02, . . . , along the 
surface to the left of 0, so that the old 
• lettering applies again. 

We now produce the lines A\^ • • • ? 

to intersection Wj, • • • ? with the arc gn 
(it is obvious that the top slope, 0(7, should 
best be drawn, in the first instance, through 
for accurately fixing the positions of 
. . .) ; then lay off, below the horizontal, 
the angle dgm — cf >; and from m, the inter¬ 
section of gm with the semicircle dAb, lay 
off the arcs mm^, , equal to 

gn^^ . . . : so that the lines gm^^ g'mg, 
. . . , all make angles equal to cfj with the 
normals to the corresponding planes Al^ 
A2 


) • • • 




55 


Next, on drawing through , 

lines parallel to the assumed direction of 
the thrust on ^ 0 , to intersection with the 
corresponding lines gm^^ g'tn^, • • • ? the 
greatest of the intercepts (g^t^ nearly) 
represents, to the scale of loads, the thrust 
on the plane ^40 ; and this length should 
exactly equal four times the length ^ 
representing the thrust from the right, as 
we find to be the case. The plane of 
rupture to the left of the vertical through 
A thus coincides nearly with A5, which is 
marked “limit” on the drawing. [On a 
larger drawing, for (/> = 33° 42' and the top 
sloping at 25°, the limiting plane was found 
to make an angle of 15° to 16° (see a more 
accurate determination in Art. 41) to the 
left of the vertical Ag, and to lie slightly 
below A5, as this drawing would indicate.] 
If we lay off along the lines parallel to 
top slope, through < 7 ^, g.^^ • • • 5 tbe true 
thrusts, ( 7 ‘ 2 ^ 2 ^ * * • ? * • * ? the 

directions of gt^, gt.^^ • • • 1 gi^^ . . . , of 

the true thrusts on the planes Al, ^ 12 , 

. . . , A7, . . . , all necessaril}^ lie above 

the first assumed directions; so that the 


56 


actual thrusts on all planes other than A5 
(which we shall regard as the plane of 
rupture, for convenience), lying above or 
below Ad, make less angles with the 
normals to those planes than the angle of 
friction, just as we found in Art. 25. 

The conditions of stability of Art. 21 are 
thus satisfied in the present case; but it is 
evident that this is no longer so if we lower 
the direction of the thrust on ^0, which 
lessens the horizontal component of the 
thrust from the right, since intersections 
like 6', 7', in the right diagram move towards 
the vertical A(/, though the reverse obtains 
for the diagram to the left, which of itself 
indicates some absurdity. If, now, we 
combine the new thrust on AO from the 
right (which has a less horizontal compo¬ 
nent than before) with the wedges of earth 
lying to the left of AO, it is readily seen 
that the directions of some of the resultants, 
as . . . , will fall below their first 
positions, and will thus make greater angles 
with the normals to their planes than the 
laws of stability will admit of ; so that any 
lowering of the first assumed position, 


parallel to the top slope, of the thrust on 
^0, is impossible. 

We thus reduce to an absurdity every 
other case but the one assumed, wliich is 
therefore true; so that the proposition 
enunciated at the beginning of this article 
is demonstrated. 

We see, therefore, that we cannot, as 
before, assume the direction of the thrust 
on the wall, AU^ as having the direction 
making the angle with the normal 
to AC\ and find the wedge of maximum 
thrust corresponding; but that its true 
direction, is found by combining the 
thrust found on ^10, acting parallel to top 
slope, with the weight of the wedge of 
earth, 0^(7, between the wall and the 
vertical plane AO ; otherwise, if the left 
diagram is constructed, we find its direction 
and amount in a similar manner to that 
used in finding the direction, etc., of 
. . . , by lajdng off on gA (produced if 
necessary) OC X 4 ; from the end of this 
line we draw a parallel to the top slope 0^ 
to intersection with the vertical through 
The line ^ to the last scale used mul- 


58 


tiplied by ^ep (where p is the perpendicular 
from A on 0(7 to the scale used in laying off 
QC) gives the thrust E against the wall in 
pounds. It is laid off in position by drawing 
a line parallel to gt^ through a point on AU^ 
^AC above A^ as previously enunciated. 

29 ((/)'< 4,). In case this construction 
gives a thrust on the wall which makes a 
greater angle with its normal than the 
co-efficient of friction, of wall on earth, 
(j)' being less than (^, then it is correct to 
assume the direction of E as making this 
angle with the normal, and proceed as in 
Fig. 4 to find the thrust. In the preceding 
article, no trial-thrust on the vertical plane 
^0 was assumed to lie nearer the horizontal 
than the top slope, as there was no reason 
for considering such exceptions to the usual 
direction in a mass of unlimited extent. 
Now, however, the wall requires the thrust 
on AO to lie nearer the horizontal than OU 
does, in which case the horizontal component 
will be increased (since intersections like 
7', 8', move away from the vertical Ag)^ 
and the thrusts on all planes A\^ A2^ • • • ? 
lying to the left of Ag^ will be raised above 



59 


their previous positions, gf^, . . . ; so 
that the thrusts on all the plniies now 
make less angles than 0 with the normals 
to those planes, so that the conditions 
for stability of ‘‘the granular mass” are 
assured. 

30. The “limiting plane,” corresponding 
to the plane of rupture on the left, can be 
found by a different construction from that 
given above. Thus, having found the line 
^ representing the maximum thrust from 
the earth to the right of ^0, multiply by 4, 
say, and combine with the successive wedges 
of earth lying to the left of AO, on magni¬ 
fying the lines 01, 02, . . . , in the same 
proportion, thus giving the lines gt^, gt^, 
. . . , for the direction of the thrusts on 
the planes ^1, ^2, . . . ; these all lie above 
the directions gin^, gm^, . . . , making the 
angles 0 with the normals to the planes, 
except for the limiting plane, where gt^ and; 
gm^ nearly coincide, as they should exactly 
if Ab was the limiting plane. The lowest 
relative position of gt with respect to giii is, 
of course, the one selected. It is evident, 
though, that the construction for the wedgo 



60 


of greatest thrust to the left of Ag gives 
a more accurate evaluation of the thrust 
than the one to the right; so that we can 
preferably use the left construction not 
only for getting the limiting plane, but for 
finding the thrust on any wall lying below 
the limiting plane. 

It is evident, from what precedes, that 
the double construction of Art. 28 applies 
only when the thrust on AO is parallel to 
the top slope ; for the moment it is lowered, 
there results several planes of rupture to 
the left of AO, which is impossible. Even 
if we attempt the left construction, we have 
seen besides that the resulting thrust on AO 
is greater than by the construction on the 
right. 

In case the face of the wall, AC, lic^s 
above the “limiting plane,” as found 
before, we evaluate the thrust on it, as in 
Fig. 4, by assuming its direction to make 
an angle with the normal equal to or to 
when <!>' < (j). Thus, if the inner face of 
the wall had the position A2, to the left of 
AO, the direction of the thrust on it would 
now be in place of gt^ as before, ami 


61 


the conditions of stability of the granular 
mass will be found to be everywhere verified' 
as in Fig. 4 (see Art. 25). 

31. Summari/. — For all cases of top 
slope, when the inner face of the wall is- 
battered, we first find the limiting plane by 
the construction of Art. 28 ; then when the 
inner face of the wall makes a less angle 
with the vertical than the limiting plane does 
(as is nearly always the case in practice, 
unless the surface of the earth slopes at or 
near the angle of repose, in which case the 
limiting plane is at or very near the vertical), 
we assume the direction of the thrust on it 
as making the angle (f) or cj)' (for cf)' < </>) with 
the normal, and proceed as in Art. 23, et 
seq.; but, if the face of the wall lies below 
the limiting plane, we proceed as in Art. 28, 
or if (f)' < (f) we may have to proceed as in 
Art. 29, to find the true thrust. 

If the wall leans backward, there is no 
need to find the limiting plane, as the usual 
construction applies. 

For earth level at top, the limiting plane 
is inclined to the left of the vertical equally 
with the plane of rupture to the right; as 


62 


the top slope increases, it)approaches the 
vertical, and coincides with it for the surface 
sloping at the angle of repose. 

Remark. — It is found from the con¬ 
struction to the right of Ag., in Fig. 5, for 
planes of rupture lying 7° to 14° above 
the one corresponding to the greatest thrust, 
that the thrust is less only by from 6 to 16 
per cent, though it more rapidly diminishes 
as the assumed plane of rupture nears the 
vertical. It must not be inferred, then, 
particularly for steep surface slopes, that 
a considerable divergence between the 
theoretical and actual surfaces of rupture 
will invalidate the theory, if the object is 
simply to get the thrust within a few per 
cent of the truth, particularly as the theory 
neglects cohesion. In fact, for a surface 
slope equal to the angle of repose, the plane 
of rupture is parallel to the surface ; but a 
plane lying much nearer the vertical will 
give nearly the same thrust. 

f 

32. In this connection, it may be well to describe 
an experiment made by Lieut.-Col. Aude in 1848, 
and repeated subsequently by Gen. Ardant, M. 
Curie, and M. Gobin, on a peculiar retaining-wall 


63 


made of a triangular block or frame, in which 
the inner face was inclined to the horizontal at the 
angle of repose of the sand backing, when, of 
course, by the usual assumption as to E making 
an angle of (p with the normal to the wall, the 
direction of E would be vertical, and there should 
be no horizontal thrust! This seemed, to the 
French experimenters, to offer a puzzling objection 
to theory; but the solution is clearly as indicated 
in Art. 28 (see Art. 67, Exps. 9 and 10). Scheffler 
indicated the correct solution as far back as 1857, 
but gave the wrong reason for it; viz., that the 
horizontal thrust w’as thereby greater. 

The writer, in “Van Nostrand’s Magazine ” for 
February, 1882 (p. 99), pointed out that any other 
solution than that indicated in Art. 28 was 
inconsistent with the stability of a granular mass, 
and the computations upon that basis agreed very 
closely with the experiments. Later M. Boussinesq 
has developed the theory of the limiting plane in 
connection with the attempt to complete the 
Rankine theory, by considering the influence of 
the wall on the pressures even to a finite distance 
from it. According to Flamant, he defines two 
limits to the thrust, and considers the most probable 
value the smaller of these limits augmented by 
^ 2 - of their difference. From an examination 
of the numerical values computed by Flamant 
(“Annales des Fonts et Chausses,” April, 1885), 
the results do not differ greatly from those given 
by the simple theory alone used in this work. 


64 


33. The disturbing influence of the wall ii^ 
changing the normal character of the stresses can 
be illustrated as follows: If the thrust on the 
vertical plane AQ (Fig. 5) acts parallel to the 
surface 0 — 10, it meets the plane of rupture at 
one-third of its length above A, through which 
point the weight of the prism of rupture acts 
also; so that the resultant on this plane acts at 
this point, which corresponds to a pressure on 
the plane of rupture uniformly increasing from the 
surface downwards. If, however, the wall causes 

C J0§§ ) 

the thrust on ^ to make a < ( angle with 

( grG3/t/6r ’ 

the horizontal, the resultant on the corresponding 


plane of rupture on the right acts 


( below \ 
\ above ^ 


the 


point situated at one-third of the length of the 
plane above A, so that the pressures on it are no 
longer uniformly increasing. This abnormal state, 
doubtless, does not extend far into the mass before 
the usual direction of the thrust in a large mass of 
earth is attained; but the fact throws doubt upon 
the assumption of a lAane surface of rupture for 
all cases where the direction of the thrust on the 
vertical plane does not act parallel to the upper 
surface. 

It appears reasonable to suppose, if the line 
through the centres of pressure on all sections 
of a retaining-wall passes through their centres 
of gravity, that no rotation of the wall occurs; 
further, if it was possible for the masonry and 



65 


earth or rock backing to settle together the same 
amount, — the backing, say, having been carefully 
deposited in horizontal layers,—then, even fora 
level-topped bank, the maximum thrust, as given 
by Rankine’s formula, will be exerted, and there 
will be no friction at the back of the wall to change 
the usual direction of the earth-thrust in a large 
bank. If the wall has not the stability, or the 
settling is not as assumed, the top will move over, 
friction at the back of the wall is exerted, and the 
horizontal thrust becomes smaller than before, 
corresponding to a different prism of thrust, as 
we ascertain by the construction of Fig. 4, for 
the two cases of E horizontal and E inclined 
downwards. The excess of the horizontal thrust 
in one case over that in the other must necessarily 
be resisted by the ground-surface, on which the 
filling rests by friction, which it is genera'lly 
capable of doing. If not, then the Rankine 
thrust will be exerted. Similarly, if we consider 
any road embankment, whose sides slope at the 
angle of repose, the horizontal thrust on some 
longitudinal plane in the interior must be finally 
resisted by the ground to one side under the 
embankment. If, however, the weight of earth 
above, multiplied by the co-efficient of friction of 
earth on ground-surface, is less than the horizontal 
thrust, the earth must slide, and the slope become 
flatter, until equilibrium obtains from a less 
horizontal thrust. Scheffier computes for an 
embankment of triangular section where f = 45°, 


( 


66 


and the angle of friction on the ground-surface is 
only 5°, that the slope of the embankment would 
change to 32° 15'. For the ground-friction angle 
= 7° 7' 20" there would be exact equilibrium; so 
that, generally, there need be no fear from spreading 
of embankments due to this cause, as the amount 
of friction required is very small. 

34. We have now given methods for 
finding the thrust against a retaining-wall, 
which simply resists this active thrust of the 
earth, for the usual cases of a surcharged 
wall and earth-level at the top, to which 
may be added the case of earth sloping 
downwards from the top of the wall to the 
rear, for which the construction is evident. 
It now remains to find the imssive resistance 
of the earth to sliding iij) some inclined 'plane 
due to an active thrust of the icall from left 
to right (Fig. 4), caused by w^ater, earth, 
or any other agency acting against the wall 
on the left. Now (Art. 22) we lay off the 
angle hgs (Fig. 4) above 6^, and then, from 
the new position of .s*. lay off arcs sSg, 
. . . , below s equal to r/rq, ga^^ . . . , as 
before, giving the direction of gs.^^ • • • » 
inclined at angle ft above the normal to the 


67 




corresponding planes Al, A2, . . . The 
construction then proceeds as before, only 
it is now the least of the resistances, pc, that 
represents the ijassive resistance of the 
earth to sliding up the plane of rupture 
corresponding ; for any increase over this 
causes the thrust on some planes to make 
greater angles than with the normals, as 
is easily shown. Let us call N the com¬ 
ponent normal to AC of the resistance and 
suppose a slight movement of the wall hori¬ 
zontally to the right; then since the earth 
moves upwards along the plane of rupture 
and the plane AC, the friction of the earth 
along AC, N tan acts upwards and the 
resistance of the wall dowmwards. 

The thrust E is now inclined at the angle 
^ above the normal to AC and not below as 
formerly. The active thrust is of course the 
only one exerted, unless the wall tends to slide, 
so that the consideration of the passive resist¬ 
ance is of small practical value. In case of a 
heavy structure resting on a foundation, we 
can replace the total weight by that of earth, 
and estimate the acti\"e thrust exerted egainst 


68 


% 


a vertical plane just below the foundation, 
for the full weight of the supposed earth, 
by the method to be given in the next 
article. The earth to one side of this 
vertical plane can be conceived to exert a 
passive thrust, w^hich may be estimated as 
explained, and should exceed the active 
thrust for a stable foundation. This 
method, though, of estimating the stability 
of a foundation, while doubtless on the 
safe side, is otherwise illusory, as any one 
who has seen a heav}- locomotive move at 
great speed along a narrow embankment 
must admit. The mass, by its friction, 
rapidly and safely transmits and distributes 
the weight over the ground, without exerting 
any horizontal thrust at the side slopes, 
which are perfectly stable. 

35. Underground Pressures. — To find 
the unit pressure at a depth x below the 
surface of a large mass of earth, level at 
top, of indefinite extent, and resting upon 
a uniformly compressible foundation, every" 
where at the same depth (see Art. 9), we 
proceed as follows : Let Fig. 6 represent a 
slice of the earth contained between two 


69 


vertical planes one unit apart, and bounded 
on one side by the horizontal plane 0(7, at 
a depth x below the surface, on the left 
by the vertical plane ^10, whose depth is 



Ax, and below by the plane AC \ the 
planes ^10, 0(7, and AIJ being supposed 
perpendicular to the plane of the paper. 
Let the length M = Ax, and the length 
be = 71. Ax. The plane AXJ will be con- 




















70 


sidered to take successively the positions 
Al<, A2, • • • ; so that if we divide 
^0 = Ao; into ten equal parts, as shown, 
and lay off similar ecpial parts along OC, 
as AC varies in position, n will take the 
successive values 0.1, 0.2, . . . Calling 
e the weight per cubic foot of earth, the 
weiglit of the prism of eartli resting verti¬ 
cally over 0(7 is represented generally by 
e.x.n. A.t, which, being directly proportional 
to w, we can lay off on the vertical OA the 
lengths 01, 02, . . . , to represent the suc¬ 
cessive values of n, or the vertical loads 
sustained by the horizontal bases 01, 02, 
. . . , of the successive prisms considered. 
When the length Ax is very small, we can 
neglect the weight of the small prism of 
thrust, xl0(7, in comparison with the weight 
of the vertical prism above it, without 
appreciable error, and ultimately find the 
position of the plane AC\ which gives the 
true thrust against ^0, by previous methods. 

Thus, draw the quadrants shown with A 
and 0 as centres, and ^0 as a radius ; note 
the intersections a^, a^’, . . . , of the lines 
-41, A2^ • • • ? with the arc 01)', next. 




71 


construct angle Cos = the angle of 
repose of the earth, and arcs ss^ = Oa,, 
ss^ = Oa.^, . . . ; so that each of the lines 
Osj, 0^21 • • • ? next drawn, make the angle 
<j) with the normals to the corresponding 
planes A\^ A2^ • • • ? find thus represent 
the direction of the resistances offered by 
these planes in turn regarded as planes of 
rupture. On drawing horizontals through 
the points of division 1,2,. . . , on AO to 
intersection 1", 2", . . . , with the cor¬ 
responding directions OSj, 0 ^ 2 , . . . , we 
note, that, if the thrust on AO is taken as 
horizontal (Art. 9), the lines 11", 22", 
. . . , represent the horizontal thrusts 
caused by the weights resting on the suc¬ 
cessive prisms AOl, A02, . . . , treated as 
successive wedges of rupture. The greatest 
of these 7T" represents the actual thrust on 
AO ; for if we assert that any other, as 44", 
represents the actual thrust, to get the 
corresponding thrusts on all the planes Al, 
A'2^ . . . , in direction and amount, we 
must lay off a length equal to 44" along 
each of the horizontals 11", 22", . . . , 
produced if necessary, and through the 



72 


extremities draw lines to 0, which thus 
represent in amount and direction the 
thrusts on the corresponding planes. But 
since 44" is less than 77", this construction 
will give a thrust on the plane A7, lying 
below the position 07^, and thus making a 
greater angle than </> with its normal, which 
is inconsistent with the laws of stability of 
a granular mass. Hence, any other thrust 
than the maximum, as given by the above 
construction, is impossible. 

The length of 77^ to scale is 0.52, which 
we must now multiply by ex Ax to get the 
total horizontal thrust on the plane in 
pounds. On dividing this thrust (0.52 
ex Ax') by the area pressed = 1 x Aaj, we 
get the unit pressure on a vertical plane 
at a depth x below the surface equal to 
0.52e.x, which is called “the intensity of 
pressure,” at a depth x. As we neglected 
the weight of the prism AOO, we must 
conceive Ax to diminish indefinitely, so that 
the error tends indefinitely towards zero, 
and the approximate intensity of pressure 
on AO = Ax approaches indefinitely that 
at the point 0. 






73 


By analysis we shall show hereafter that 
the plane of rupture, A1 in this case, bisects 
the angle between the natural slope and the 
vertical. 

In this case we have taken 0 = 18° 26% and the 
resulting intensity (0.52ea:) is found to be exactly 

that given by the usual formula, ex tan^ 

The intensity at any point of a vertical plane thus 
varies directly with x. The total amount on a 
vertical plane of depth x from the surface is then 
x Cx^ 

Cxdx = - 7 — (where C = 0.52e in the present 

case), and its resultant is at a depth z equal to the 
limit of the sum of the moments of the pressures 
{Cxdx) on the elementary areas dx X 1, taken about 
the top surface,‘divided by the total pressure, or 

^ ~Jo • 2 ~ 

Also, ^ X 0.52x = ^ X line represent- 

2 2 2 ' 

ing thrust, if old construction is used. These 

are precisely the conclusions derived from previous 

constructions. 




In case the top surface is sloping, a 
similar construction applies, only OC must 
now be drawn parallel to the top slope, and 
the pressure on OA must be assumed to act 






74 


parallel to this direction. The construction 
is similar to that given for Fig. 5 (on 
neglecting the weight of the wedge of 
thrust as above), either to the right or left 
of the vertical Ag^ only as the weight of 
the prisms vertically above 01, 02, . . . . 
(Fig. 5) is now represented by ex nAxy. 
cos i (where i is the inclination of the top 
slope to the horizontal), we must multiply 
the length of the line cc' (Fig. 5) to scale, 
by ex cos .to get the intensity of the 
pressure at the depth x^ since the lengths n 
alone were laid off to represent the loads 
gg 2 . . . , as in Fig. 6, and the resulting 
thrust cc' must now be magnified ex ax cos 
i times to get the thrust in pounds on the 
plane ax X 1. As ax approaches zero 
indefinitely, the approximate intensity 
ex AX cos i — 

--cc', on the area ax y 1, ap¬ 
proaches that at the depth x (=ex cos i. cc') 
as near as we please. It must be observed 
that AO in Fig. 5 must be taken equal to 
unity in this construction, and the same 
scale used in laying off the distances along 
the top slope 0 — 10. 





75 


36. If the earth to the right of AO, in 
Fig. 6, does not experience the similar 
active thrust of earth to the left of AO, hut 
only the passive resistance of a tunnel 
lining, etc., of an underground structure, 
the conditions are changed if this lining 
gives in consequence of its elasticity; for 
the wedge of thrust, AOC, cannot move 
to the left without developing friction along 
the surface 0(7, therefore the pressure on 
this surface must no longer he taken as 
vertical, but as inclined at a direction 
0—10', making an angle with the vertical 
(Fig. 6). The load on any supposed wedge 
of thrust, as A04:, is now represented by 
04', the thrust on AO by 4'4", and the 
pressure on the plane A4: by 04". The 
greatest of the lines, I'l", 2'2", . . . , now 
represents the true thrust, and it is readily 
found to be 4'4" = .33 to scale; so that the 
intensity of the thrust on a square foot at 
the depth x is now 0.33e^, or one-third the 
intensity on the horizontal plane 0(7. Mr. 
Baker Science Series,” No. 56) found 
for a heading, driven for the Campden-Hill 
Tunnel, at a depth of 44 feet from the 






76 


surface,—tbe angle of repose of the over- 
lying clay, sand, and ballast, heavily 
charged with water, being only 18° 26' as 
assumed above,—that the relative deflec¬ 
tions of the timbering in the roof and sides 
indicated that the vertical and horizontal 
intensities of pressure were in the ratio of 
3.5 to 1, which is very near what we 
obtain by the last construction. The first 
construction indicates a ratio of onlv 2 
to 1. 

In most cases, a portion of the weight of 
the earth above the tunnel is transferred 
to the sides (Art. 9), though here it was 
thought that the full weight of the 
ground took effect upon the settings.’' 

We have now carefully examined the 
conditions of interior equilibrium of a mass 
of earth, and ascertained the thrusts ex¬ 
erted, whether in the interior or against a 
retaining-wall; and we see that the graphi¬ 
cal method is capable of handling, with 
equal ease, any case that ordinarily pre¬ 
sents itself. The results, of course, agree 
with the analytical method, founded on the 
same hypotheses j but as it is often more 


77 


convenient to calculate the thrust, even 
when a graphical method is afterwards 
used for testing the stability of the wall, 
we shall now proceed to deduce formulas 
for evaluating it. 


78 


CHAPTER III. 

THEORY OF RETAINING-WALLS. 

Analytical Method. 

37. As in the preceding chapter, we 
shall assume a plane surface of rupture, 
and regard the mass as subject only to the 
laws of gravity and frictional stability 
stated in Art. 21. 

In Fig. 7, let AFP QUO represent a 
cross-section of the earth-filling, taken at 
right angles to the inner face of the Tvall 
AF. We shall consider the conditions of 
equilibrium of a prism of this earth con¬ 
tained between two parallel planes, per¬ 
pendicular to the inner face of the wall, 
and one unit apart, regarding the wall 
AE as resisting the tendency of the earth 
to slide down some plane, as AC, passing 
through its inner toe. 

Call G the weight of the prism of earth 


/ 


79 

AF'PQliC in pounds, directed vertically j 
Ey the earth-thrust against the wall AF, 
directed at an angle of friction of earth 
on wall when or of when 

Trig, -r 



below the normal to the inner face of the 
wall (Art. 7)5 and S the reaction of the 
plane A C, inclined at an angle <f> (the angle 
of repose of earth) below the correspond¬ 
ing normal, since the prism is supposed to 
be on the point of moving dozvn the plane' 





A C. These three forces are in equilibrium 
when E and jS act towards 0 and G acts 
downwards. 

Call the angle that AC makes with the 
horizontal and the angle FAC, ft. On 
drawing the parallelogram of forces as 
shown, we have, since E and G are pro¬ 
portional to the sines of the opposite 
angles in the triangle ONE, 

E sin ONL 

G sin NLO 

It is easily seen from the figure that ONL = 
y — and that NLO — ^ ft “f“- J 
hence the above general relation becomes, 

E _ sin {y — <j>) ^ ^ 

G sin {cf) -y cf)' -y ft) 

Now, if we conceive the plane AC, always 
passing through the point A, to vary its 
position, that value of E, corresponding to 
the greatest value obtained by the con¬ 
struction above, is the thrust actually ex¬ 
erted against the wall; for, if JL (7 is the 
plane of rupture corresponding to this 
greatest trial thrust, any less value of the 




81 


resistance o£ tlie wall E will cause S to 
make an angle greater tkan ^ with the 
normal to AC, which (Art. 21) is inconsist¬ 
ent with the law o£ stability o£ a granular 
mass (also see Art. 25): hence the least 
thrust consistent with equilibrium corre¬ 
sponds to the greatest value o£ E thus ob¬ 
tained 5 and this is the actual active thrust 
exerted against the wall, when the wall 
simply resists the tendency to overturning 
or sliding on its base, caused by the ten¬ 
dency o£ the prism o£ rupture to descend. 
I£ there is a thrust exerted on the wall to¬ 
wards the earth, £rom any external £orce 
acting on the le£t o£ the wall j £rom le£t to 
right 5 then, i£ this be supposed to increase 
gradually, the active thrust o£ the earth on 
the right is first overcome; then, as the ex¬ 
ternal £orce increases, the directions o£ S, 
on all planes as AC, approach the normals 
to those planes, pass them, and finally tlie 
£ull 2 )cissive resistance o£ some prism of earth 
to sliding upwards along its base is brought 
into play. The greatest force E, as regards 
sliding up the base of some prism, which 
can be exerted is that corresponding to the 


least o£ tlie trial forces, E, obtained by 
supposing tbe position of the j^lane JL(7 
to vary, for S lying above tlie normal to 
^ C at an angle 4* feacli plane 5 for if we 
suppose A C to represent tlie corresponding 
plane of rupture, if tlie external force, 
equal to E, and acting from left to riglit, is 
increased, it necessarily causes tlie direc¬ 
tion of S to make a greater angle than 4> 
witli the corresponding normal, which is 
inconsistent with equilibrium (Art. 21). 

In this chapter we shall only consider 
the passive resistance of the wall to over¬ 
turning or sliding caused by the active 
thrust of the earth tending to descend, 
which is all that is required in estimating 
the stability of retaining-walls. 

38. We shall now express the value of* 
G for the earth-profile shown in Fig. 8 ^ 
taken to represent the general case, and 
proceed to find the maximum value of E, 
for different trial-planes, which represents 
the actual thrust exerted against a stable 
wall. We shall suppose the true plane of 
rupture to intersect the part EY of the 
profile j the line IIY is then produced to 


83 


jB, so that the area of the triangle ABC 
is equal to that of the polygon AFPQRCj 
which can be effected by ordinary geomet¬ 
rical means. The point B therefore does 
not change, as we suppose the position of 
C to vary between B, and T. 



Let us drop the perpendicular AT from 
A upon B Y, and designating by e the 
weight per cubic foot of earth, we have 
G = he.AT.BC. 

For future convenience we have desig¬ 
nated, in Fig. 8, the angle that AC makes 
with the vertical u?, and the angle that the 
inner face of the wall AF makes with the 









84 


vertical so that the angle P of (1) is 
now replaced by (cs? -f- if the wall leans 
forward, or by {od — a) if the wall leans 
backwards. 

In Fig. 8, let us draw the line Cl, mak¬ 
ing the angle ACT = cf)' p) — 

4- ^ 4" intersection I, with 

the line of natural slope AD through A, 
If the wall leans backwards, 

^ CJ =: (<^ -f- + OD — a). 

Since the angle {y — <P)— CAI, we can 
replace the ratio of'the sines in (1) by that 
of the sides opposite in the triangle A Cl, 
or of Cl to AI; so that, substituting the 
above value of G, we can write (1) in the 
following form:— 

E = ie.AT.BC.A^. . . (2). 

-A JL 

On drawing BO parallel to Cl to inter¬ 
section 0 with AD, we have, from this 
relation and the similar triangles, BOD 
^nd CID. 

BC= BD ; Cl = ID. — : 

OD OD' 




85 


which substituted in (2) gives, 






OD- 


y 


OI.ID 



The terms in the ( ) remain constant as 
we vary the position o£ A C. For brevity, 
call AI — X, AD ~ a, AO = h ; then we 
can write the variable term, 

OI.ID _ {x — h) {a — x) _ I ; ab 

AI X ^ X ’ 


which is a maximum for x = as we 

find by placing its first derivative equal to 
zero. This value of x substituted in the 
variable term gives. 


a -j- b — 2^/ab 


{a — \/ab)^-y 
a 


so that the actual thrust E on the wall can 
be written,— 

. . . (4). 

\ / a 


Now, drawing the perpendiculars BN and 
CII from B and C upon AD^ we observe 
that since the angle A CH = go ^ 
makes the angle go with a vertical at C, 










86 


and CH makes the angle with this same 
vertical, since the sides are respectively per¬ 
pendicular to those o£ the angle DAJ—<f>), 
and the whole angle = 
it follows that the angle IICl — NBO = 
3-8 marked, if the wall leans for¬ 
wards j otherwise IICI=]SfBO= {(f>' —or), 
since ACI'i^ then equal to (ce?-|-</)q-</)'—or), 
as previously observed. 

To reduce (4) to a simpler form, we 
remark that AT.BD represents double the 
area of the triangle ABB, and can be re¬ 
placed by AB.BN = AB.BO cos OBN; 
which gives 

AT.BD.BO 
- - --= a cos 

OJD^ 

.•.E = ie. cos OBN (‘ 

(a — ^dbY . . . (5). 

Now, from similar triangles, BOB, CIB, 
BO Cl 

we have =■ __, which, substituted in 


OBn(—\ ; 
\0I>} ’ 

IZ»/ 


the above expression, Ave have, noting that 

{a — /y/ ah) ~ {a — oc) = IB, the very sim¬ 
ple formula, 





87 


E=zhe. cos {^' -\-a) CE . . . ( 6 ). 

It is to be remarked, that, if the wall leans 
backwards, cos {(f)' a) is to be replaced 
in this formula by cos ((f>' — a) 5 further, if 
we lay off IL = IC on the line Zd., and 
draw a line from L to C, the thrust E is 
exactly represented by the area of the 
triangle ICL multiplied by c, the weight 
per cubic foot of the earth. 

39. This simple conclusion has been 
previously reached, in an entirely different 
manner, by Weyrauch (see an Nostrand’s 
Magazine” for April, 1880, p. 270), who 
states that Rebhahn in 1871 found a similar 
result, assuming, however, that — 0 . or 
<fi' zzz (f) (for the special cases of earth-level 
at top, or sloping at the angle of repose, I 
infer). 

Recurring now to the fact, that for the 
true plane of rupture we found 

X = AI =1 ah — y\/AD.A 0, 

and that angle NBO =(<^'-f-^) c>r — nr), 
according as the wall leans forwards or 
backwards, we have the following simple 
construction to find the plane of rupture 




88 


and earth-thrust E, as given by Weyrauch 
in 1878, for a uniform slope and wall lean¬ 
ing forward. 

Having found the point JB on the pro¬ 
longation of the line BY, which it is thought 
will be intersected by the plane of rupture, 
so that area ABB area AFPQB^ we 
next draw BO, making the angle NBO 
with the normal to the line of natural slope 
AD, equal to (</>' + or {<f)' — a), accord¬ 
ing as the inner face of the wall lies to the 
left or to the right of the vertical through 
A (replace ^^by (j> whenever j then 

erect a perpendicular at 0 to AD to inter¬ 
section M, with the semicircle described 
upon HD as a diameter, and lay off AI = 
chord AM, since AI = VH O.AD / next, 
draw IC parallel to OB to intersection C 
with the top slope, whence A C Avill be the 
plane of rupture if the point C falls upon 
J?Yas assumed; otherwise another plane, 
as YZ, will have to be assumed as con¬ 
taining the point C, and the construction 
effected as before. 

Having found C in this manner, E can 
be computed from (1), since 0 = ^ AT.BG 



89 


is now known: or by measuring Cl to 
scale, E can be found directly from (6). 

This graphical construction is more 
rapid and accurate in working than the 
methods of the preceding chapter, and is 
superior to Poncelet’s construction, in 
taking less space to effect. 

In surcharged walls, the point B will 
generally lie to the right of AF. Thus, in 
Fig. 4 the upper line 26 is extended to the 
left 5 from 0 a line is then drawn parallel 
to A2 to intersection 0' with the line 26 
extended. The point 0' thus found corre¬ 
sponds to the point B of Fig. 8. 

40. The construction is true whether the 
earth-surface slopes upwards or down¬ 
wards from the top of the wall. 

In the latter case, if the surface, say i>7), 
falls upon the line BO, the construction 
fails ] but a formula given farther on gives 
the value of E. 

If the surface BD falls below BO, it is 
easily seen, on drawing a figure, that all the 
previous equations hold, and we reach the 
same conclusion as before, AI= ^AD.A 0 j 
only as ^0 now is larger than AD, the 



90 


semicircle must be described upon JL 0 as 
a diameter, and tbe perpendicular to tbe 
point M erected at D ; or AI can be calcu¬ 
lated if preferred. If tbe points 0, Ij 
and D are near together, it will be best to 

compute BG from BG ~ since 

the terms in the right member can be 
measured to scale. 

41. Position of the Limiting Plane .—In 
Fig. 9, let BI) represent the earth-surface, 
uniformly sloping at the angle i to the 
horizontal, of an unlimited mass of earth 
(Art. 9), in which the pressure on a verti¬ 
cal plane, AB, can be taken as parallel to 
the surface BD. Let AD represent the 
line of natural slope; it is required to find 
the position of the plane of rupture A (7, 
corresponding to the thrust F7, acting 
above the horizontal at the angle i, and of 
course balancing the opposed thrust of 
the earth to the left of AB. 

On referring to Fig. 7, it is seen that 
equation (1) holds on replacing the de¬ 
nominator of the right member by sin 
(/?-|-^—/). Therefore, in Fig. 8, the angle 


91 


ACI must now be laid off equal to 
?), wlience the line Cl falls below 
CH, and BO below BN, botli being in¬ 
clined to these normals at the same angle, 
= i-|-0= i. 

With this exception, the above demon¬ 
stration holds throughout, and we reach 



the following construction to find the point 
C. From B draw BO, making the angle i 
below the normal BN to AD, or preferably 
making the angle (</> — i) with the vertical 
AB, to intersection 0 with AD. From 0 
draw OJI perpendicular to AD to intersec¬ 
tion 31, with the semicircle described upon 





92 


AD as a diameter; lay off AI along ADj 
equal to chord AM^ and from I draw a 
parallel to BO io intersection (7 with'the 
top slope BD. The plane (7 is the plane 
of rupture, or the limiting plane of Art. 28, 
which see. 

If the inner face of the wall lies below 
AC^ then (Art. 28) the thrust=^e. cos i. CD 
on AB is computed, and, regarded as 
acting parallel to BD, from left to right, 
is combined with the weight of the earth 
and wall to the right of AB to find the 
true resultant on the base of the wall. 

If the wall lies between AB and A C, the 
constructions of Arts. 37 and 38 are used. 

To be as accurate as possible in these, as 
in all constructions, true straight edges on 
both ruler and triangle are imperative. 
Lay off all angles, including right angles, 
by aid of a beam compass to a large 
radius, say ten inches, using a table of 
chords (except for the right angle) and an 
accurate linear scale. With all care, the 
angles BAC thus found can scarcely be 
counted on to nearer than ten minutes, 
which, however, is sufficiently accurate. 



93 


In tlie table below will be found, for 
various inclinations i, the values of the 
angle BA C that the limiting plane makes 
with the vertical; also the co-efficient K 
(see Art. 42), or the thrust on AB—^e 
cos i when AB and e are both taken 
as unity, made out for earth which naturally 
takes a slope of one and a half to one, or 
whose angle of repose is 33° 42'. 

The value of K agrees fairly well with 
calculation, the last figure not differing 
more than one or two, at the outside, from 
the results of Art. 47. 

From the construction we see that as i ap¬ 
proaches ^ indefinitely, BA C tends to zero 
and E approaches the limit cos <^. AB-^ as 
given by analysis. The increase of thrust 
is very rapid from i=30° to «=<^ = 33° 42'. 


i 

0 

0 

o 

o 

o 

15° 

0 

o 

25° 

26°34 

0 

o 

CO 

33° 42' 

BAC 

28°09' 2G° 

24° 

21°50' 

19° 10' 

10° 

14° 40' 

0 

o 

0° 

E 

.143 '.145 

1 

.149 

.157 

.172 

.194 

.207 

.244 

.416 



































94 


42. Uniform Top Slope; Formula for 
FartJi-thrust —When the upper surface of 
the earth slopes uniformly at the angle i to 
the horizontal, it is easy to deduce from 
what precedes a general formula for the 
thrust exerted by it. Fig. 10 represents a 



retaining-wall leaning towards the earth. 
We shall first deduce a formula for this 
case, when it will be observed, as we pro¬ 
ceed, that the same formula holds, when 
the wall leans forward, on simply changing 
to ( — af 

In this case, we note from Fig. 10 the 
following values for the angles.— 






95 


NBO = (f>' — 

ABO=cf>-{- cf>'^ 

AOB—90— 

ABB ~ <f> — 

ABB = 90 H" 4- 
BAG 90—{<i> +a). 

Finally, designate by I the length AB from 
the inner toe to where the inner face of the 
wall intersects the top slope, and by li its 
corresponding vertical projection. 

From formula (5) we deduce, remem¬ 
bering that OB {a — ?>), 

E = ie.WO^- P — cos OBN... (7). 

La — 6 J 


We can now write the [ ] as follows :— 




Place n = 



to find its value in terms 


of the functions of known angles, we have 
from the triangles AOB and ABB by the 
law of sines. 







96 


AO __ sin (<^ + 4>') _ sin {<j> — i) - 

AB cos — «')’ AD cos -j- i)' 


On multiplying these two equations to¬ 
gether, and extracting the square root, we 
find, 

_ I AO _ j sin {sl>^^') sin (0 —t) 

^ \l AD \ cos (9!>'— a) cos (a + t) ’ ' 

Again, from the triangle we have, 


B0= 


cos {cf> -j- a) 
cos {(f)' — a) 


I - 


Substituting these values in (7), and putting 
cos OBN = cos (cj)' — a) for this case, and 
we have finally. 


^ _/ cos {4>-^a) Y 
\ +1 / I 


eP 


2 cos {(f)" — a) 


(9). 


Or, since h = l cos u', we likewise have. 


/cos {<ji -f a)\^ eli^ 
\(w+l) cos a! 2 cos {4>' — o') 


( 10 ). 


If we term the co-efficient of elfi in (10), 
we can write this formula, 

E = Kelfi . . . (11) 














97- 


in which, for tvalls leaning hachivards, as in 
Fig. 10, 


^ _ /c os (c^ + 1 

V(n-|-1) cos a; 2 cos (</>' — a) 



where n has the value given in (8). 

For walls leaning forwards, we easily 
note the changes in the angles used, and 
can verify that formula (11) obtains j but 
now. 


f'cos (c^ — or)\2 1 

<(w-f-l) cos aj 2 cos {<!>' d- a) 



and. 


n = 



sin -f- </>') sin {d> — i) 
cos {di' 4- a) cos {a — i) 



which we obtain from the old values by 
simply changing a io {— oc). 

It is to be observed, for all cases, when 
<i>' > cf> that we must replace 4^' in all the 
formulas by </>. 

These formulae are identical with those of 
Bresse (^^Cours de M4canique Appliqu4e,” 
Vol. I. 3d ed.) and Weyrauch, for the 
case of the wall leaning forward, the only 
cases examined by them. Bresse uses the 









98 


Poncelet method for the general case, 
which leads to Poncelet’s celebrated con¬ 
struction. The routes pursued by these 
authors is different from that given above, 
the method of Weyrauch, in particular, 
being much more complicated 5 still, all 
three methods lead to precisely the same 
formula, so that it must be considered as 
established beyond question. 

Weyrauch, too, in subsequent reductions, 
follows Rankine as to the direction of the 
earth-thrust against the wall, whereas 
Bresse takes it as above. The case of the 
^^limiting plane’’ is not considered by either. 

43. The case where the top surface slopes 

downwards to the rear is very rarely met 
with in practice. The previous formulae 
apply though directly on simply changing i 
to (— i), since it is seen that angle ADB 
= -f- i) and angle ABJD = 90-|- {a —i), 

and the ratio —— is now equal to —^ 

Al) cos(n:— i) 

44. Earth Level at Top / Back of Wall 
Vertical .—For the earth level at top, back 
of wall vertical, and (f)' — ^ as usually 
taken, the formula ( 11 ) takes a very simple 



99 


form. Here we have a=0^ i=0j 

whence. 


J sin 26 sin 6 . , 

-^^ = sin V2, 


cos 


and 




cos cf> 


.eJi^ . . . (15). 


2(1+ sin V2)^ 

For <fi' ~ 0, which corresponds to a per¬ 
fectly smooth tvall, or otherwise may refer 
to the direction of the pressure on a ver¬ 
tical plane in a mass of earth of indefinite 
extent, level on top (Art. 9), we have, 
when = 0 and i =0, n z=z sin ^ and, 

1 — sin 4* 


E = 


1 + sin 4> 2 

= tan2 ^45° — . 


eli^ 


. . . (16). 


The equality of the two co-efficients of 

« 7,2 

—- in (16) is easily verified from the known 

2 


formula. 


tan^ . (*) = * 


cos X 


by putting (90 — <^) for x in both members. 
Referring to Fig. 7, and regarding AF 











vertical, the top surface horizontal, and 
z= 0, we note that tan § and 


E — tan (5 tan {y — ^), in which y = 
2 


90 — fi. Now, this result must agree with 
the right member of (16), which is only- 


possible when ^ J 45 



(90—; whence it follows that for ^'=0, 
a: = 0, i =: 0, as assumed, the plane of rup¬ 
ture bisects the angle between the vertical and 
the line of natural slope. 

45. Earth sloping at the Angle of Repose .— 
For this case we shall assume a= 0 and 
<f}' — <f) in addition to ^ whence n=0 


and, 


E 


cos <t> 


eh^ 


( 17 ), 


as found in a different manner in Art. 41. 
This simple formula can likewise be de¬ 
duced directly from equation (1) of Art. 
37, referring to Fig. 7, 

sin [y — <h) 


E 


cos (/i 4- cf>) 





101 


On substituting the value of G, which is 
easily found for this case to be, 

^_ 1 eh- __ sin (3 cos eh- 

cot (3 — tan <h 2 cos [(3 2 ^ 

we find for the trial thrust. 

^_ sin /? cos eh- 

~ sin (2 </) + p) 2 

_ cos (j) eh^ 

sin 2 cot f3 -[- cos 2cf> 2 

. Now, by the reasoning of Art. 37, the 
true thrust is the greatest value the above 
expression can have, as (3 varies, and its 
greatest value corresponds to /5=90— 
for then cot (3 is least, and JE greatest, since 
cot (3 is ill the denominator. On substi¬ 
tuting this value a simple reduction gives 
E — h cos 3) . eh^ as found above in (17). 
Since we have just found, for this case, 
that p = 90 — it follows that the surface 
of rupture coincides with the natural slope. 
The value of E from equation (1) in this 
case assumes the form 0 X oo, since G 
becomes infinite for an indefinitely sloping 
surface; but on reducing to the form above 








102 


we easily see the limit that E approaches ' 
indefinitely, which is its true value. The 
construction of xVrt. 39 fails for this case, 
but the one of Art. 41 leads directly to (17). 

46. Pressure of Fluids. —The general 
formula (9) above is true, no matter how 
small the angle of repose becomes, and 
must approach indefinitely the expression 
for the pressure of liquids, as and 
tend towards zero; so that at the limit, 
for — cj)' = i = 0, we have the normal 
thrust of a liquid whose weight per cubic 
foot is e, 

E — \ eP cos or = 4 eh- see a . . . (18), 
a well-known formula. By Art. 44 we see 
that for (f> — 0, 2/3 = 90, or the plane of 
rupture approaches an inclination of 45° 
as <f> approaches zero indefinitely. 

47. PanJcine’s Formula for the Earth-thrust 
on a Vertical Plane, in an Indefinite Mass, 
sloping uniformly. In Art. 9 we have 
stated the conditions that such a mass 
must satisfy in order that the pressure on 
a vertical plane, whose intersection wdth 
the top slope is a horizontal line, may be 
parallel to the line of greatest declivity. 


103 


Also in Art. 28 we liave seen, that, when 
the wall face lies below the limiting plane, 
this direction of the thrust is the true one 
on a vertical plane, passing through the 
inner toe of the wall. 

We have a= 0, <f>' — i. and I = h, which 
gives in formula (9), 


where, 


E = 


/cos eli^ 
\n -f-1^ 2 cos i 


n 


_ I sin (<^ + i) sin (</> — i) 

cos- i 


_ a/ sin^ cos^ i — cos^ 4* sin'-^ i 


cos i 


_ V cos^ i — cos'-^ 4> 


cos ^ 


Whence, 
E = 


cos^ <f) cos i 


_ ej^ 

(cos i -\- '\/ cos- i — cos^ ^ 

Now, since we can write, 

cos2 4> (cos i + Vcos2 i _ cos" </)) 

X (cos i cos^ i — cos^ </>) 


the above value becomes, on striking out 















104 


tlie common factor, (cos i+Vcos'^ i _cos'^ 


= Vcos^ i—cos2 <f> 


COSl 


cos i-\- cos*-^ i —cos^ (j> 


, ( 19 ), 


which is Rankine’s well-known formula 
for earth pressure. 

Now since Rankine’s formula was framed 
without the use of any assumption, as that 
of a plane of rupture, and is accepted as 
correct for the case in question, it follows, 
that, when the pressure is assumed to be 
parallel to the surface, the assumption that 
the surface of rupture is a plane will give 
correct results, and can be safely used in 
the graphical method which is absolutely 
dependent on it. 

It will be observed that formulae (16) 
and (17) can be deduced directly from (19) 
by making ^ = 0 and i = cf) respectively. 
Rankine has given a simple graphical con¬ 
struction of the last fraction in (19) in his 
“ Civil Engineering,’’ which saves labor in 
computing. 

48. U7iit Pressures on a Vertical Plane 
at Dep)tli X heloiv a’ uniformly Sloping Sur¬ 
face, the Direction of the Pressure being 






105 


taken Parallel to the Line of Greatest De¬ 
clivity. —As in Art. 35 we shall consider a 
wedge of thrust of infinitesimal dimen¬ 
sions, of which the left face AB (Fig. 10) 
is vertical, and the upper surface paral¬ 
lel to the top slope. The weight of 
the vertical prism that rests upon any 
trial base as BC is, e . BC . cos i .x .= 
AT . BC .exi AB (Fig. 8); so that neglecting 
the weight of the infinitely small wedge 
ABC we get the value of E from equa¬ 
tion (1) of Art. 37 by simply replacing G 
by this value. Equation (2) of Art. 38 is 
thus replaced by 

E=^.AT.BC% 

which is exactly that given in Art. 38 mul¬ 
tiplied by the constant 2x1 AB. All the 
subsequent reductions, therefore, hold if we 
simply put h =AB in the final equations, 
and multiply the result by 2x1 AB. Hence 
divide (19) by AB = h and change the 
coefficient eh 12 to ex, to find the intensity 
of the pressure, E ^ AB, at a depth x; 
and on integrating this expression, mul¬ 
tiplied by dx, between the limits o and h, 





106 


we are at once conducted to (19), which 
is thus proved true by the method of 
integration of the effects of earth particles, 
which is independent of the assumption of 
a plane surface of '’upture extending to 
the surface. 

Precisely the samu conclusions hold for a 
vertical ivallj or one leaning fonvards^ when E 
is assumed to make the angle cf)' or <h with the 
normal to the wall, since G is simply replaced 
as before by the weight of the vertical 
prism for a uniform top slope, and ultimately 
we replace Ifi by 2x in the general formula 
(11) to get the intensity of pressure in the 
direction given, at the depth x from the 
surface, so that on integrating as before we 
deduce (11) without the necessity of con¬ 
sidering the surface of rupture as extend¬ 
ing to the surface. The graphical method, 
using this hypothesis, should again give 
good results. It is possible though, in this 
case, that the influence of the wall friction 
may have some effect in deflecting the 
weights of the vertical prisms from a ver¬ 
tical line 5 for, when it is so transmitted, the 
usual direction of the pressure is parallel 


107 


to tlie surface (Art. 9). For ivdlls leaning- 
hackwards the prisms do not rest vertically 
over the bases of the prisms of thrust,* and 
the theory would seem to be inapplicable ; 
so that the formulae for this case, (8) and 
(9), have to rest upon the unproved hypo¬ 
thesis of a jilane surface of rupture extend¬ 
ing to the surface, and may depart consid¬ 
erably from the truth. We conclude, that,, 
except for the cases for which Rankine’s 
formula is applicable, the plane surface of 
rupture is still an unproved hypothesis. 

49. Pointof Application of the Thrust; Uni¬ 
form Slope .—We have the normal compo¬ 
nent of the thrust on a wall, by (9) whether 

the wall inclines forward or backward or 
is vertical, expressed by the relation, 

= (9) X cos </)' = 

c being constant j whence the thrust on 
the area X 1 is nearly 

dE-^ = 2cldl, 

and the distance from where the inner 
face of the wall interesects the top slope 
to the centre of pressure is equal to 
the limit of the sum of the elementary 


108 


pressures multiplied by their arms, divided 
by the total pressure, or, 



hence the centre of pressure on the wall is 
vertically above the base, 

50. Surcharge uniformly distributed .—If 
the filling of height h has a horizontal 
surface upon which a uniform load of any 
kind rests, replace its weight by that of 
an equivalent quantity of earth, giving the 
total load the same, and call the height of 
the reduced load /d.. The total pressure on 
the vertical wall of height h is now by (11), 
E = Ke {{h + /d)2 — 7 / 2 ) ^ xeh (h + 2/d), 
whence, 

dE = Ke.2-{h + W) dh ; 

and the distance of the centre of pressure 
from the top of the wall downwards is, 

2 j ^ (h + h') Mh ^ ^ 

h (h + 2!7Y^ 3 \ h + 2/i' / ’ 

or from the base of the wall upwards, 





109 


9/^2 3 /^/^. 

^ Sh + 6h' 



It is more tlian probable that the theory 
for this case will prove illusory in practice, 
and will give a large excess of pressure; 
so that, most frequently, such surcharged 
loads are ordinarily allowed for by a large 
factor of safety, particularly where the 
earth is bound by cross-ties, stringers, etc., * 
or the surcharge is not free to move later- » 
ally as well as vertically. 


Surcharge 



/ I 

L ^ 


I ^^2 


c 

ICioj. lO (a). 


In the case of sea tvaUs, the backing is 
saturated with water at high tide, uj) to a 
certain level BF, fig. 10 (a), so that it is 
well to ignore the friction at the back of 










110 




the wall on BC and for additional safety 
it will be neglected on the portion AB. 

Call the weight of the backing per cubic 
foot above BF^ e-i and the angle of repose 
The corresponding quantities for the 
saturated backing below BF will be desig¬ 
nated by 62 and (j> 2 - The value of ^2 
should be found by experiment and 62 
computed as explained at p. 31. If the 
water, at high tide, is at the same height 
on the front and back faces of the wall, 
the water pressures on those faces will 
balance and need not be considered. 

Let and earth thrust on^^j 

BC—I 12 and i^ 2 =earth thrust on BC. 


By Art. 44 (IG), K — \ Tan^ ^45°—.^^ ; 

therefore if — TF= surcharge or load in 

pounds per square foot, on AB; by the 
analysis just given, 

= A'r.A, (7ii + 27(>) = taii^ ^45“ —p\ 

_l_ above S, a 

distance, ^ 


(i+ 


\7ii 

h, + 2h') 3 : 



Ill 


Next, assuming that the load on the hori¬ 
zontal plane BF is uniform and Wo lbs. 
pr. sq. ft. and calling the height of this 
load reduced to the specific gravity of the 
earth below BF, Jiq, 

.’. ho 62 = Wo W -|- hi eu 


Hence as before, 

E2 = tan2 (45=- — + feWo] 

and E 2 acts above (7, a distance, 



ho \ 

h2+2}J 


b 

3 


If no surcharge is considered, the formu¬ 
las apply on making, 

W=o, h'—o . *. Wo—hi ejj ho—hi 61—62. 


In‘either case having found and E 2 in 
magnitude and position, the position of 
the resultant Ei -{- E 2 ^ can be found by 
taking moments about C. 

The above formulas will be found to 
reduce to those given by Mr. D. C. Berber, 
in Engineering News, Aug. 23, 1906. It 




112 


is stated there, that the Department of 
Docks of New York City specify a sur¬ 
charge of 1,000 lbs. pr. sq. ft., acting as a 
vertical load. 

51. Moments of the Thrust about the Inner 
Toe of the Wall .—Let us decompose the 
thrust E against the wall into two compo¬ 
nents, E\ and E 2 ., respectively normal to 
and acting along the inner face of the wall. 
If E makes the angle with the normal 
to the wall, we have, from E = Keli^, 

El = E cos <f>' = K cos (fi'.eh^ • 

or putting, Ki = K cos (f>' 

we have, Ei = Ki eli ^; 

also, E 2 — E sin 4 *' = Ei tan 

It is understood in these formulae, that, 
when > <^, we must replace 4 >' by 
If the inner face of the wall makes an 
angle a with the vertical, we have the 
thrust acting at a distance cl = eh sec a 
from the inner toe of the wall, where c = 
by theory for a uniform slope ; there¬ 
fore, the moment M of the thrust about the 
inner toe of the wall is Ei cl, since the mo- 


113 


ment of E 2 is zero; or putting for abbre¬ 
viation, 

m ■=■ c K\ sec a 

we have, 

M — E\ cli sec a — c Ki sec a . eh^ — meW. 

The two formulas for the normal compo¬ 
nent of the thrust and moment, E\ — Kieli^ 
and M=meh^ will be used throughout the 
next chapter, as semi-empirical formula?, 
in which the constants, c, and m are to 
be determined by experiment; and it is 
well to recall that h represents the vertical 
height from the inner toe of the wall to 
where the line of the inner face pierces 
the top surface of the earth backing, and 
that e represents the weight per cubic foot 
of earth. 


114 


CHAPTER IV. 

EXPERIMENTAL METHODS. — COMPARISON 
WITH THEORY. 

The Practical Designing of Petaining-Walls. 

52. Experiments upon earth-pressure 
have been made by a number o£ engineers 
and others at various times, but none have 
ai^proached in completeness those made 
by M. M. L. Leygue, and described in 

Annales des Pouts et Chaussees ” for 
November, 1885. From them, in fact, can 
be deduced, for the materials used, a com¬ 
plete method for the practical designing of 
retaining-walls, as will be more fully appre¬ 
ciated as we proceed. We only have space 
to indicate the more important results, 
which are given briefly below. 

53. Cohesion .—Leygue first proved that 
cohesion in a mass of sand does exist, but 
that for this material it can often be nejr- 

) 


115 


lected in comparison with friction. Thus 
for a vertical mass about a foot high, and 
ten square feet base, the ratio of the resist¬ 



ance of cohesion to friction, to sliding on 
the base, is only one-tenth. 

54. The Prism of Rupture .—To observe 
the movement of the sand at the moment 
























116 


o£ sliding or overturning of the retaining- 
wall, a wooden box was used (Fig. 11), 1.31 
feet long, 2.62 feet wide, and 2.4 feet high, 
open above and in front, but with glass 
sides. Movable boards in front, pivoted 
at the bottom, represent the retaining- 
walls. They were 1.22 to 1.3 feet long, 
and 0.66 to 0.82 feet high, and were first 
put at the inclination desired, when the 
sahd (whose natural slope was one and a 
half base to one rise) was placed in the 
box behind them. To note the relative 
movement of the grains of sand, as the 
wall moves over, thin horizontal strata of 
%vliite plaster pulverized were interposed in 
the mass of gray sand at various heights. 

In all the experiments, narrow strips of 
paper, bent at a right angle, were placed, 
over the joints to keep out grains of sand, 
and allow of free overturning or sliding. 

As the retaining-board was moved over, 
the following facts were observed:— 

All of tJie particles lying hetiveen the wall 
and a convex surface (shown by a dotted 
line in tlie figure) moved parallel to this 
surface, which may be called the surface of 


• 117 


rupture. This curve was determined by 
noting where the thin layers of plaster be¬ 
gan to break, which was easily done ; it 
remained invariable as the rotation of the 
wall was increased, even up to 30° for a 
horizontal surface of earth; but on com¬ 
pleting the rotation by lowering the retain- 
ing-board to the horizontal, the curve 
moved slightly to the rear, and the earth 
finally took its natural slope. In this first 
movement, the particles, moving all parallel 
to the surface of rupture, had velocities 
proportional to their distances from this 
surface. In fact, since the mass to the left 
of the curved surface was invariable during 
a part of the rotation, it follows that the 
areas of the triangular bases remained 
constant, and that their vertices (the in¬ 
tersection of the surface of earth with the 
wall) must move parallel to the curved 
surface of rupture. The full friction of the 
earth teas exerted against the wall, so much 
so, in fact, as to raise the earth next the 
wall slightly, and thereby diminish its 
density. 

The convex surface of rupture, roughly 


118 . 


speaking, bisects the distance from where 
the upper surface meets the wall, to where 
the horizontal through the last point meets 
the line of natural slope through the foot 
of the wall; thus for high surcharges 
differing essentially from theory. 

The same conclusions were found to 
hold when the retaining-wall was moved 
parallel to itself, except that the curve of 
disjunction tended from the start to move 
towards the interior of the mass. 

• Similar conclusions were found to hold 
for millet seed or for large rocks as back¬ 
ing ; and the surfaces of the bases 'of the 
prisms of rupture were found to be similar 
figures for heights varying from 0.66 to 
6.56 feet so that their areas can be sup¬ 
posed to vary directly as the squares of 
their heights in accordance with theory. 

The angle of friction of earth on the 
rough wooden board was found to be 
<f)' — 39°, for earth on glass <f>' = 24° 30' • 
and yet, when a glass wall was used, the 
surfaces of the mass of rupture were in¬ 
creased only about six per cent., or they 
were practically the same. 


These surfaces for millet seed, whose 
natural slope was two base to one rise, 
were thirty-seven per cent, greater than 
for sand with a horizontal top surface, and 
about double for a surcharge sloping at 
two to one. The following table gives the 
values of x and y (Fig. 11) for li = 1 for 
different inclinations of the surcharge i to 
the horizontal, and inclinations a of the 
wall to the vertical, counted positive where 
the wall is to the right of the vertical or 
leans towards the earth, and negative 
when to the left 5 also the greatest dej)ar- 
ture/of the curve of disjunction from its 
chord, and the areas of the bases of the 
prisms of rupture, all referring to sand 
whose angle of repose was = 33°42'^ 
or a slope of I 2 to 1 :— 


120 




Table No. 1. 


tan i 

tan a 

y 

X 

/ 

.Vreas of Bases 
of Prisms 
of Rupture. 

0 

1 

— ^ 

0 

0.552 

0.088 

0.394 


0 

0 

0.662 

0.053 

.290 


+ 5 

0 

0.786 

0.031 

.201 



0 

0.858 

0.022 

.161 


2 

3 

0 

0.933 

0.015 

.127 

1 

1 

3 

0.885 

1.437 

0.120 

0.853 


0 

0.662 

1.324 

0.077 

.558 



0.453 

1.239 

0.047 

.322 


1 

2 

0.358 

1.216 

0.033 

.236 


2. 

3 

0.267 

1.200 

0. 

.172 

2. 

3 

1 

3 

1.770 

2.322 

0.150 

1.338 


0 

1.324 

1.986 

0.096 

.818 


+ i 

1.161 

1.692 

0.059 

.454 



0.1)06 

1.574 

0.040 

.319 


2 

3 

0.716 

1.437 

0. 

.220 


For any value of li as 3, we find tlie cor¬ 
responding values X, and f by multiply¬ 
ing the tabular 'values by 3, and for the 
areas by multiplying by 3^ = 9. Interme¬ 
diate values are easily obtained by inter¬ 
polation. 

55. Surcharge uniformly distributed .—If 
we conceive a vertical wall backed by 
earth level with its top, of height h, with a 
further surcharge of earth vertically over 


















121 


the first of a uniform height h', and pre¬ 
vented from spreading by the sides of the 
box, and a movable board in front in pro¬ 
longation of the first retaining-wall, we 
have the conditions of the experiment. 
As the lower wall was rotated, the earth 
escaped from under the other, and the sur¬ 
face of rupture, which still passed through 
the foot of the lower wall, was no longer 
found to be a simple convex curve. The 
lower part was as before j but near the top 
of the lower wall it deflected nearly verti¬ 
cally upwards for a short distance, and 
then again towards the interior, thus pre¬ 
senting two points of inflection, and giving 
a much less surface of rupture than theory 
would seem to demand. It is true, though, 
that much of the weight of the surcharge 
was held by friction by the upper wall, so 
that theory no longer applies. (See exper¬ 
iments recorded in ^^Engineering News” 
for May 15 and 29,1886, giving, by weigh¬ 
ing, the exact weight of sand held up by 
the sides of a box when the bottom was 
gradually lowered.) 

56. Values of the Moment m as obtained by 


122 


Experiment .—To ascertain the moment of 
the earth-thrust against the foot of the 
inner face of the retaining-hoard (Fig. 11), 
the force applied at the upper end of the 
frame constituting the extension of the 
board, and perpendicular to its direction, 
that was necessary for exact equilibrium, 
as overturning was just about to begin, 
was measured by the tension of a rope 
passing over one or two pulleys, and 
weighted at the further end by a weight F. 
The weight of the board and frame, when 
inclined, was balanced by the pull of a rope 
at top. Allowing for the friction of the 
pulleys, as found by experiment the resist¬ 
ing force applied at the top of the frame, 
2.46 feet from the toe of the wall, was 1.05 
F on an average, and its moment about 
the toe was l.OS F' X 2.46. This must 
balance the moment of the earth-thrust, 
which, for a length pressed of 1.3 feet, 
would be, if the sides of the box did not 
cause friction, mel^ x 1.3 (Art. 51), whence 
m can be found when A", e and li are given. 
But the lateral pressure of the sand 
against the sides of the box causes fric- 


123 


tion, and tlius hinders the forward motion 
of the sand, and thus diminishes the 
thrust. To estimate its influence, Leygue 
follows Darwin’s plan of measuring the 
force F' (corresponding to .F) when a par¬ 
tition-board of thickness t is placed per¬ 
pendicular to the wall, and centrally in the 
mass. As the retaining-board does not 
receive the full pressure on its length, 1.3 
feet, which call d, we can suppose, for no 
central board, the thrust exerted on a less 
length \d = {d — 2x) to correspond to a 
full thrust if the sides exerted no influ¬ 
ence. Here A. is a co-efficient of contrac¬ 
tion analogous to that used in the case of 
liquids, and x represents the total contrac¬ 
tion caused by one side of the box. The ratio 
of the forces F and F' will, therefore, be, 
F ^ d — 2x 

d—t — 4:X 

since the full thrust, when the centre 
board is used, acts only on the length 
(d — t — 4ix) as the influence of the fric¬ 
tion of four side walls has now to be con¬ 
sidered. The forces F and F' were 
directly measured, and the values of x 



124 


found from the last equation, which, sub¬ 
stituted in the preceding equation, gave 
the co-efficients of contraction A. For 
level-topped earth 1 — A was found to 
amount to 5^, and for the surface sloping 
at the angle of repose to as much as 15^, 
which illustrates the marked influence of 
counterforts in diminishing the actual 
thrust exerted against the face wall. The 
results can be represented by the formula, 

A = 0.135 (7 — tan i) j 

hence the true value of m, corresponding 
to the full thrust if there were no side 
walls, is to be found from the equation, 
1.05 F X 2.46 

= 1.3 X meh^ X 0.135 (7 — tan i). 

57. The values of F, as found by exper¬ 
iment, corresponding to h = 0.66 feet or 
less, and e = S9 pounds per cubic foot, 
enable us to compute the values of m given 
in the next table. 

The values of m for millet-seed, whose 
angle of repose is 26“ 34', or tan 
are likewise inserted, being obtained by 
experiment in the way indicated above. 


125 


Table No. 2. 


e 

a 

Values 

or 9n FOB Sand. 

Values of m fob 
Millet-Seed. 

ci 

tan i= 0. 

tan i= 

tan i=? 3 . 

tan 0. 

tan i= X- 

_3 

*> 

0.535 

0.970 

1.329 



_T 

T 

.241 

.425 

.574 

— 

— 

_2. 

O 

.152 

.260 

.356 

— 

— 

1 

3 

.003 

.103 

.136 

0.084 

0.154 

0 

.030 

.047 

.005 

.057 

.106 

1 

3 

.015 

.024 

.032 

.037 

.063 

2 

3 

.000 

.014 

.018 

— 

— 

a 

¥ 

0 

0 

0 

— 

— 


58. To these most valuable observations 
are to be added those o£ Mr. George Dar¬ 
win, Trinity College, Cambridge, made 
in a box 0.72 feet long, 1.8 feet high, and 
1 foot wide, after the manner explained 
above. 

Vertical wall, hacking in horizontal 
layers not compacted. . . . = 0.030 

Do. compact filling . . . . ilf = .022 
Vertical wall, backing sloping back¬ 
wards at 3 to 2.= .0275e/t® 

Do. layers sloping upwards at 3 to 2, = . 0315^/4 ® 

Mean for loose earth .... Jf = 0.030 

The co-efficient 0.030 is the same as that 



















126 


given above for level-topped earth; but for a 
surcharge sloping at the angle of repose 33° 
42', Mr. Darwin found m — 0.048, whereas 
Leygue’s constant = 0.065. Leygue sug¬ 
gests that this difference is to be accounted 
for from the small size of the box, which 
caused the prism of thrust to be truncated. 

It is doubtful whether Leygue^s constant 
is great enough for a mass sloping indefi¬ 
nitely, for the prism of rupture for earth at 
the natural slope was barely contained in 
his box. Still, theory does not indicate a 
marked increase in the thrust for very high 
surcharges, over those about equal to the 
height of wall; but we have seen in Art. 41 
that theory gives, for the surface sloping at 
the angle of repose, 33° 42', a thrust nearly 
double that corresponding to a surface 
slope of only 7° less in place of the ratio 
65 to 47 given in the table. It is just here . 
that we should expect, and will find, the 
greatest divergencies between theory and 
experiment; for cohesion is much more 
marked for a long surface of rupture than 
for a short one, which doubtless causes the 
surface of rupture in the experiments to 


127 


lie much nearer the vertical than theory 
would indicate. This divergence becomes 
still more marked for walls leaning back¬ 
wards, as we shall see farther on. 

59. The experiments of M. Gobin (1883) 
were of the same character as those of 
Darwin and Leygue, though not so exten¬ 
sive as those of the latter. Leygue’s re¬ 
sults are about 14^ inferior to Gobin’s, 

’ which Leygue asserts is due to the stiffness 
of the cords used, and the large (20^ of 
the load) friction resistance of the pulleys, 
the stiffness of the hinges used at the foot 
of the board, manner of filling in, and 
lastly the appreciation of the instant of 
sensible movement. 

60. Direct Estimation of the Normal Com- 
portent of the Earth-Thrust^ and of its Point 
of Application —To estimate the normal 
component of the thrust, the same box 
(Fig. 11) was usedj but the single board 
was re^Dlaced by a frame containing two 
parallel boards separated by springs, whose 
compression gave a direct measure of the 
thrust. The normal component of the 
weight of the inner movable panel was 


128 


equilibrated by the pull o£ a rope passing 
over a pulley when the panel was inclined 
to the rear, and directly measured by the 
springs, and allowed for when it was in¬ 
clined forward. The movable panel was 
likewise equilibrated for the friction caused 
by the rubbing of the earth in its double 
movement of sinking and translation, by 
the pull of ropes acting upwards in the 
plane of the j^anel, which thus prevented 
the scraping of the panel on the bottom of 
the box. Direct experiment showed that 
the force necessary to slide the panel up¬ 
wards, when the full earth-thrust was 
exerted against it, was about equal to the 
normal component Ei multiplied by the 
co-efficient of friction f — tan as has 
been hitherto assumed; so that, to get the 
resultant pressure on any wall, we must 
combine the value of found bv aid of 
the following table with E^ tan </>, which 
thus gives the direction of the pressure as 
inclined below the normal to the wall at 
the angle 4*. The method of experiment¬ 
ing was to fill in behind the movable panel, 
equilibrated as explained, and kept tempo- 


129 


rarily at tlie same distance from the fixed 
panel by a fastening j then the fastening 
was suddenly removed, and the earth sud¬ 
denly exerted its thrust, so that the com¬ 
pression of the springs following was double 
that due to the constant thrust of the 
earth. Calling half the force measured by 
the springs on a length c?, and for the 
vertical projection of the area pressed 
equal to h X d^ we have, admitting the 
law of the parabola (Art. 51), 

= K, elfi\d, 

where A, as before, is a co-efficient of con¬ 
traction. The value of A. was determined 
as explained in Art. 57, by experiments 
with the spring apparatus, to be A := 0.120 
(7 — tan i). 

From these formul^e the value of was 
obtained as given in the following table. 

• By multiplying by sec we obtain the 

co-efficient K in the formula for the total 

earth thrust, E = Kelfi, on a wall one foot 
»• ' 

long. 

By taking the foot of the movable panel 
as a centre of moments, we have the 


130 


moments o£ the forces recorded by the 
springs equal to the moment of the double 
earth-thrust, from which the point of ap¬ 
plication of the latter can be readily found. 
Calling the fraction of the height from the 
base to the point of application of the earth- 
thrust c, we thus obtain the quantities in 
the column headed c in the next table. 


Table No. 3. 


e 

Pi 

Spring Appa¬ 
ratus. 

c 

Spring Appa¬ 
ratus. 

A'l 

• Aja 












tan i= 0 



tan i= 0 

X 


tan 0 

X 

% 

_3 

2 

0 418 

0.461 

0.485 

0.712 

1.223 

1.519 




_1 

T 

.432 

.467 

.486 

.390 

.645 

.836 




O 

3 

.439 

.467 

.485 

.243 

.367 

.508 





.437 

.462 

.480 

.136 

.211 

.271 

0.165 

0.271 

0.366 

0 

.425 

.448 

.470 

.071 

.104 

.137 

.094 

.147 

.188 


.401 

.429 

.455 

.038 

.054 

.067 

.048 

.071 

.090 

3 

.383 

.411 

.438 

.019 

.029 

.034 

.023 

.032 

.040 


61. It seemed to the writer that it would 
be of interest and value to use the areas 
of the bases of the prisms of rupture 
(Art. 54) to represent G in Fig. 3, Art. 21, 

























I 


131 

and by the method indicated there to find E 
and E^ = Ki^ when e = \ and h =1. 

The actual surfaces given in Leygue^s 
tables were used, and the constructions 
effected on his drawings, using the chord of 
the line of rupture as giving the average 
direction of the corresponding surfaces of 
rupture. 

The values thus found of this normal 
component, which is termed are given 
in the table just to the right of the values of 
Ki as given by the spring apparatus for 
comparison, and it will be observed that 
they are uniformly larger than the values of 
K^. To find the distance to the point of 
application of the thrust for 1i—l, we 
have to divide m of the table of Art. 57, by 
JiTj® sec n', and the results are given in the 
following table. As it is of value in this 
connection, the same thing was done for 
millet-seed, where tan </) = 26°34'j 

the moments being taken from the table. 
Art. 57, and the thrusts /Pi being found 
by taking the volumes of the prisms of rap¬ 
ture as given by experiment, and finding 
E^ from the construction shown in Fig. 3. 


132 


Table No. 4. 




For Sand. 

Millett-Seed. 

BY 

tan a 

tan i 

_ m cos a 

a m cos a 

Theory 



^ Kia 

K^a 

c 

1 

3 

0 

0.363 

0.381 

1 

3 

1 

3 


.360 

.363 

i 

3 

1 

3 

A 

3 

.352 


1 

3 

0 

0 

.319 

.420 

J 

3 

0 

1 

.326 

.450 

1 

3 

0 

A 

3 

,346 


J 

3 

1 

3 

0 

.296 

.389 

1 

3 

1 

3 


.321 

.427 

1 

3 

1 

3 

2 

3 

.337 


1 

3 

•A 

3 

0 

.325 


1 

3 

f 


,366 


J 

3 

2 

3 

1 

.375 


1 

3 

Average. 

0.340 

0.405 



Since the actual wedges o£ rupture, by 
experiment, vary as the squares o£ the 
heights, should equal one third, £rom 
which it does not differ very greatly tor 
sand. We should conclude £rom the value 
o£ c given by the apparatus with springs, 
that the point o£ application o£ the thrust 
was considerably above the i h given by 
theorv,lying generally between 0.4 and 0.5, 
which last value it can never equal. But^ 


















133 


taking the thrusts due to the actual wedges 
of rupture, the values are much nearer the 
theoretical. Neither method is exact, since 
in the first case, as the sand advanced, it 
rubbed on the bottom, and thus hindered 
the advance there more than at the top; 
and, in the second case, the wedge of rup¬ 
ture cannot act as an invariable solid 
sliding down two planes at once as theory 
requires.* 

It can be said, though, in support of 
Leygue’s values of Jf, and c, that their pro¬ 
duct gives almost exactly the value of m 
(Art. 57), determined in an entirely difEer- 
ent manner, and the results are of course 
correct for a wall that has slid about the 


1 Perhaps a correct value of the horizontal thrust could 
be obtained by putting a wide, though light retaining- 
wall on wheels, and attaching cords to the bottom hori¬ 
zontally to the rear, where they could pass over pulleys, 
and weights be attached sufficient to prevent motion; 
then, after filling in the sand behind the wall, the 
weights can be diminished until motion just begins, and 
the horizontal component of the thrust read off, allowing 
for friction. The wheels could preferably be of the kind 
used on sliding doors to diminish friction as much as 
possible, as the vertical component of the thrust would 
add to it considerably, which, however, could be equili¬ 
brated as in Leygue’s experiments. 



134 


average compression o£ the springs; but it 
is possible that the initial thrust before 
the wall gives is somewhat different. 

The average of c given by Leygue^s 
spring apparatus for millet-seed was 0.417, 
or less than for sand 5 and he therefore 
concludes that c approaches its theoretical 
value as diminishes. 

62. Influence of tlie Character of the Re¬ 
taining- Wall .—When the retaining-boards 
were of glass, it was found that, on an 
average, 

1 = 1.019; ^ = 1.02 and — = 1.053, 
c Ki m 

where c', /f/, and m' refer to the quanti¬ 
ties obtained with a glass wall, and c, JTi, 
and m to the roughened wooden one. For 
the wood on sand, tan = 0.810, and for 
the glass, tan = 0.456. 

63. Surcharge imiformly distributed .—For 
the case of Art. 55, the following values 
were found with the apparatuses before 
used:— 



135 


Table No. 5. 


Height of 
Surcharge. 
h' 

Centre of 
Pressure 

C3 

K^s 

ms 

0.00 

0.425 

0.070 

0.030 

0.5h 

0.434 

0.094 

0.042 

iXh 

0.440 

0.117 

0.051 


These are much lower values than the ordi¬ 
nary theory of Art. 50 would give^ which, 
however, is hardly applicable here. 

As K varies between 0 and /i, we can 
express the values of c, Ki, and by the 
following empirical formulae:— 

c’ = c (l + 0.035^0 ; = 

+ 0.66 ; »«• = m{l + 0.7 

where c, and m refer to li — 0. 

For h — h'j the value of JE given by the 
formula of Art. 50 is about double that 
found by experiment, and the centre of 
pressure lies above in the ratio of 1.333 to 
1.035. 

64. Comparison with Theory .—In the follow- 










136 


ing table are given the theoretical values, 
side by side with those obtained from actual 
experiment, for the angle that the plane of 
rupture makes with the horizontal, and also 
the values of K and m in the formulas for 
the total thrust, JE^E^ sec 
sec (f> • (Arts. 42 and 51), and for the 
moment about the inner toe of the wall, 
M=meh^. Tan a is regarded as minus when 
the wall is inclined away from the earth, 
and plus when inclined towards the earth. 


Table No. 6. 


tan a 

tan i 

Angle of Rup¬ 
ture WITH THE 
Horizontal. 

Co-efficient 

K OF 

THE Thrust 
E^Keh^. 

Co-efficient 

m OF 

OVERTURNING 

Moment. 

M=meh^. 

Theory. 

Experi¬ 

ment. 

Theory 

Experi¬ 

ment. 

Theory 

Experi¬ 

ment 

1 

—s 

0 

60°21' 

61^10' 

0.222 

0.163 

0.065 

0.063 


1 

2 

50° 

58° 10' 

.411 

.254 

.120 

.103 


Z 

3 

3304 O' 

57° 

.750 

.325 

.232 

.136 

0 

0 

56°36' 

56°30' 

0.133 

0.085 

0.037 

0.030 



4703 O' 

54°50' 

.214 

.125 

.060 

.047 


2 

3 

33°40' 

54°10' 

.416 

.165 

.115 

.065 

1 

3 

0 

50°12' 

51°50' 

0.072 

0.046 

0.021 

0.015 


1 

9 

44°48' 

51°10' 

.109 

.065 

.032 

.024 


2. 

3 

3304 O' 

5 O 05 O' 

.218 

.082 

.064 

.032 




























137 


It will be observed that for i = 0, the 
values of the angles of rupture as given by 
experiment (average inclination of the 
curved surface of rupture was taken) agree 
well with theory, as also the values of m; 
but for surcharged walls the theoretical and 
actual values differ materially, especially 
for tan for which value for a leaning 
wall the theoretical moment is double the 
actual. This has been partly foreseen, 
since for tan the theoretical plane of 
rupture approaches indefinitely the natural 
slope in direction, which calls for an infin¬ 
itely great wedge of rupture. The theoreti¬ 
cal thrust, however, does not differ material 
lyfrom one corresponding to a plane of rup¬ 
ture lying much nearer the vertical, as has 
been hitherto mentioned j so that the differ¬ 
ence must be due to some other cause, in 
which cohesion, of course, plays some part. 

.Whatever may be thought of the values 
of e and Ah obtained by the spring appa¬ 
ratus, there can be no question about the 
accuracy of the value of wq which is of 
predominating influence where overturning 
is considered. 


138 


65. Stability of Betaining- Walls against 
Overturning. — To illustrate tlie use of 
Leygue’s constants in testing the stability 
of a retaining-wall of height li (Art. 51) by 
a graphical method, we first compute the 
values E = El sec 4)' — Ki sec elfi (put 
4> for Tvhen cf)' > and eh from the 
assumed values e and h, and the values of 
the constants c and Ki, derived from the 
table (Art. 60); then lay off to scale E 
from the inner face of the wall, at a verti¬ 
cal height equal to eh above the base, and 
making an angle below the normal to the 
inner face equal to (or </> if ^' > as in 
Fig. 3, and then combine this thrust with 
the weight of the wall W, acting along the 
vertical through its centre of gravity, to 
find the resultant on the base. Its inter¬ 
section with the base gives the centre of 
pressure on the base ; and this point should 
at least lie within the middle third of the 
base, for reasons given in Chap. I. The 
wall need not be drawn for more than half 
its height; and, if the drawing is made to 
a large scale, the results are very accurate. 
This graphical method should always be 


139 


used in testing the results as given by the 
method of moments for stability against 
overturning. 

To illustrate the use of the method of 
moments, let us consider a wall leaning 
towards the earth, the back making the 
angle a with the vertical, and suppose the 
base perpendicular to the back, and of 
thickness t. 

We may consider three forces as acting 
on the wall: the normal component 
acting at a height cli above the base, or at 
a perpendicular distance cli sec a from the 
outer toe ; the friction E^^ tan (j>' (use 4* for 
cf>' when 4>' > (f>) of the earth on the wall, 
acting downwards along the back of the 
wall at a distance t from the outer toe j 
and, lastly, the weight W of the wall, 
acting along the vertical through its centre 
of gravity, and at a horizontal distance g 
from the outer toe. 

If we call (7 the co-efficient of stability, 
or the factor by which it is necessary to 
multiply the normal component (leaving 
the friction at the back of the wall the 
same) to cause the resultant on the base 


140 


to pass througli the outer toe, we have, 
taking moments about the outer toe, 

Wg = El {(T cli sec a — tan .t) 

= Ki G c sec a eli^ — E 2 t 

. •. 11^ — a melt^ — E2t ; 

since (Art. 51), Ki c sec a 

= m and Ei tan — E 2 . 

The factor of safety, o', was not made 
to increase the total friction at the back of 
the wall, because the usual causes which 
influence and increase the thrust are pass 
ing loads and rains 5 and the latter, in lubri¬ 
cating the surfaces of contact, would prob¬ 
ably cause the total thrust on the wall to 
approach nearer the normal than before, 
so that it is not safe to consider the fric¬ 
tion increased in the same ratio as the 
normal thrust. 

The formula gives m as the controlling 
constant in the right member. If we take 
this constant from the table in Art. 57, 
where it was accurately found from experi¬ 
ments on rotating retaining-boards, the 
result should closely agree with the reality; 
for the term E-zt is of minor importance. 


\ 


141 


and in case Leygue’s values of and con¬ 
sequently of E 2 =Ei tan are too small, 
the only effect is to add to the stability of 
the wall designed by the above formula, 
and is therefore on the side of safety. 

From the above formula we can ascer¬ 
tain the factor of safety (j for a given wall, 
or for a given stability find the thickness t 
(on expressing TFand g SiS a function of t)j 
when the other quantities are assumed. 
For O'—! we have exact equilibrium about 
the outer toe, supposing no crushing there. 

See Art. 68 for the derivation of a more 
general formula when the base of the 
wall, of length is horizontal. 

For the case above to find the distance u 
from the inner toe of the wall to the centre 
of pressure on the base, call = horizontal 
distance from the inner toe to the vertical 
through the centre of gravity of the wall, 
and we have the equality of moments, 
about the centre of pressure, of W, i^i), 
and E 2 expressed as follows:— 

W {u cos a — g') = meh^ — ti -E' 2 , 
from which u can be found, and the centre 


142 


o£ pressure located. I£ u = §t, there will 
be no stress at the inner toe (Art. 17), and 
for greater values of u the joint would tend 
to open. The wall can be designed by 
this formula for any given value of ex¬ 
pressed in terms of t if preferred, 

66 . Stability of a Hetaining-Wall against 
Sliding .—The simple test in this case is, 
that the resultant on the base of the wall 
shall make, with the normal to the base, 
an angle less than the angle of friction 
between the wall and its foundation,—a 
condition that can always be satisfied by 
sufficiently tilting up the base in front, 
which safeguard should never be neglected, 
particularly for dock or river walls, which 
generally fail, if at all, by sliding. The 
angle of friction of masonry on wet clay is 
only about 18°, 

It is equally necessary to go down deep 
enough for a firm foundation, or to drive 
piles, preferably in the direction of the 
resultant on the base. The foundation- 
course is generally made larger than the 
base of the wall, to better distribute the 
pressure on the base. 


# 


143 


67. Discussion of Experiments on Betain- 
ing-IValls.- —In tlie following table is given 
a resume of quantities relating to a number 
of experimental retaining-walls all at the 
limit of stability by actual experiment. 

The eartb-tbrust against tbe various 
walls bas been computed by three differ¬ 
ent methods :— 

1st, By theory (Chaps. II. and IIL). 

2d, By using the thrust resulting from 
the actual wedge of rupture for sand as 
given in Art. 61, using value Ki. In 
both these cases the point of application 
was taken at one-third height of wall from 
the base. 

3d, By Leygue’s method, using the con¬ 
stants and c for sand — 33° 42'), 
given in Art. 60. 

From the insufficiency of our experi¬ 
mental data for all materials, varying in 
their angle of repose within practical 
limits, the same constants had to be used 
by the second and third methods, for 
varying values of (when between 33° 
and 40°), which strictly apply only to 
■= 33° 42'. 


Table No. 


144 

















































145 


For brevity we denote by,— 
li the height"of the wall in feet, 
t its thickness at the base, 
e the weight per cubic foot of earth, 
w the weight per cubic foot of wall, 
a the angle the inner face of the wall makes 


with the vertical counted | according 

{ towards» 


as the wall leans 


1 


the earth. 


(from 

i the angle with the horizontal of the top sur¬ 
face of the backing, 

4> the angle of repose of the backing, 

«#>' the angle of friction of earth on wall, 
g the ratio of the distance from the centre of 
pressure on the base of the wall to the outer 
toe, to the thickness t, using first method 
above (theory), 

g' ditto, using second method, or actual wedge 
of rupture (for sand) method, 
g " ditto, using Leygue’s or the third method, 
g, q' or q" will be taken-]- or — according as the 
resultant strikes within or without the base 
of the wall. 


The walls were all of uniform cross-section, 
except Nos. 8, 9, and 10. Walls 1, 3, 4, 8, 9, 10, 
were of wood with sand backing, except the first 
which had shingle backing. Lieut. Hope’s wall. 
No. 2, was of bricks, with ordinary earth for back¬ 
ing. Curie’s waU, No. 7, was in Portland cement, 


146 


backed by damp sand; but Nos. 8, 9, and 10 were 
peculiar triangular frames, whose inner faces made 
the angles 27° 30' and 53° with the vertical (see 
Curie’s “Poussee des Terres,” Paris, 1870, and 
“Trois notes” in 1873). In No. 8 the face exactly 
coincided with the “limiting plane” (Ai*ts. 28and 
41), and in Nos. 9 and 10 was below it; so that the 
thrust for No. 8 was found as usual, the values of 
,, and being interpolated from the tables. 
But for Nos. 9 and 10 the thrust was first found 
on the limiting planes, respectively 27° 30' and 28° 
15' to the left of the vertical, which was then com¬ 
bined with the weight of earth and frame to the 
left, assuming the thrust on the limiting plane to 
make an angle ^ with the normal, as theory de¬ 
mands. 

In wall No. 7 the surcharge extended entirely 
over the top of the wall to a height of 4.26 feet, 
and was then level. The values of g and q" were 
not found for this wall, since </> = 45° was so much 
g]’eater than 33° 42', to which q' and q" refer. In 
wall No. 1 the earth did not reach the top of the 
wall by 0.25 feet. Gen. Pasley’s wall was omitted 
from this list, as possibly the base was imperfect, 
since the theoretical force necessary to apply at 
the top to overturn it was to the actual as 20.6 to 
17.53. The value of q, however, was found to be 
-f- 0.12, and of g' .08 for A = 2.17 feet, i = 0.67 

^oet ~ = 1.06,a = i= 0°, and </> = </>'= 39°. The 
w 


147 


values of g, q\ and q" were found for some stable 
walls, not quite at the limit of stability, to be posi¬ 
tive as they should be. All of the values of 7i, 
given in the third column of the table, are simply 
averages of several values found by the experi¬ 
menters under the same conditions. 

It is stated by Flamant, that a retaining-wall 
made of a very light, empty wooden box, of a 
width slightly greater than the height, and hin¬ 
dered from sliding by a small obstacle placed at the 
exterior edge, which did not interfere with rotation, 
was just able to support the thrust of sand, filled in 
to the height of the box. This experiment has 
afforded ground for ingenious speculation, but is 
perfectly explained if we take the co-efficient of 
friction of sand on the side of the box at about one- 
third. The experiment should have been com¬ 
pleted by ascertaining the value of this co-efficient. 

It may be of interest to state that, using Ean- 
kine’s formula (16), p. 99, for the thrust ard ignor¬ 
ing, as he does, the friction at the back of the wall 
for (X = 0 ^ i = 0 , the values of q for walls 1, 2, 3, 
4, 5 of Table 7 are found to be, —0.79, —0.58, 
_0.74, -1.33, -0.33. 

An inspection of the average values of 
q, q and in the table shows that they 
are very near zero, except for walls leaning 
towards the earth, where “theory”diverges 
greatly from the reality. For vertical tvalls, 


148 


with earth level at top, either method gives 
practically the same results when cf> is near 
33° 42', to which the experimental con¬ 
stants alone apply. We can then use, with 
confidence, either method. It is likewise 
seen from experiments 11 to 14 on leaning 
walls that theory fails here, but that g' and 
g' are practically the same; so that either 
the second or third methods can be used. 
It was an agreeable surprise to find that 
the second method gave as good results as 
the third throughout. As it is a very simple 
thing to find by experiment with any earth, 
the actual wedge of rupture, the practical 
value of this discovery is evident, as it can 
then be used, as explained in the preced¬ 
ing chapters, to evaluate the thrust. 

It is a little disappointing that the third 
method, using Leygue’s constants, does not 
agree more exactly with his experimental 
retaining-walls. To point out the discrep¬ 
ancies more fully, the value of /^, for the 
assumed t = 0.1 foot, were computed by 
Leygue’s method (third), and by theory 
(first method), and compared with the 
average value given by his experiments. 


149 


Table No. 8. 

t ASSUMED AND h COMPUTED. 


No. 

Actual. 

t. 

By Leygue’s 
Method. 
h 

By 

Experiment. 

h 

By Theory. 
h 


Feet. 

Feet. 

Feet. 

Feet. 

5 

0.1 

0.443 

0.367 

0.436 

6 

0.1 

.321 

.285 

.306 

11 

0.1 

.755 

.755 

.669 

12 

0.1 

.492 

.476 

.371 

13 

0.1 

1.230 

1.310 

.961 

14 

0.1 

.692 

.738 

.456 


To judge from these experiments on the 
very small model walls, we should natur¬ 
ally infer that Leygue’s method gave the 
thrust too small for vertical walls, and too 
great for walls leaning at an inclination of 
■^5 but, doubtless, these discrepancies will 
disappear when larger walls are used. In 
fact, for a peculiar triangular wall 6.6 feet 
high, composed of rubble in a thin mask of 
wood with thin counterforts, his theory ap¬ 
plied very well j the wall, however, break¬ 
ing above the base until a wooden base was 
attached, when it moved over in one piece. 













150 


Experiments G and 7 (see previous table), 
for a surcharged vertical wall, the earth 
sloping at the angle of repose, would seem 
to indicate that theory was verified in spite 
of the fact that the constants by Leygue 
are Ah=0.137 and 0.065, whereas by 
theory/ii=: 0.346, and ^=0.115, for ^ — 
33° 42' 5 but this apparent verification is 
to an extent accidental, and resulting from 
the special relations of o' and t for these 
cases. If we assume that the masonry 
weighs one-fourth more than the earth, as 
in a following article, then for (7 = 3 and 
h = \ we find by the formula of Art. 65 
that t = 0.58 by theory, but only 0.49 by 
Leygue’s method for the surface sloping at 
the angle of repose. For the surface slop¬ 
ing at 1, theory gives t = 0.45, and Leygue 
0.424 j and for earth level at top, theory 
gives t = 0.367, and Leygue t — 0.343. 

This shows that theory agrees fairly well 
with experiment for gentle slopes, but for 
the surface sloping at the angle of repose 
it gives too large values. It is not a little 
remarkable that with the values of Jii, and 
above all of m, differing so much from 


151 


those of Leygue, for this case, that the 
computed values of t should agree so 
closely by the two methods. 

Perhaps future experiments, with higher 
surcharges, may cause the difference to 
disappear entirely. 



The above are undoubtedly the most 
valuable experiments we have, as they 
refer to walls in the ordinary conditions of 
their employment in practice, and should 
be carefully studied. 







152 


68 . General Formula for Stability of TtO' 
taining-Walls against Overturning .—Let Fig. 
12 represent a wall ABCD, whose length 
perpendicular to the plane of the paper is 
unity^ and whose exterior and interior 
faces and diagonal AC make angles with 
the vertical equal to and oo respec¬ 
tively. Let W denote the weight of the 
wall, and g the horizontal distance from its 
line of action to the outer toe A^ also call 
O' the factor by which it is necessary to 
multiply the normal thrust leaving 

the friction^ 1^0 "back of the wall 
constant, in order that the resultant on 
the base may pass through the outer 
toe. Here f =■ tan <{>' (when > <^, re¬ 
place cfj' by <f)) and the quantities h, t, d, iVj 
i, cf) and <}>' have the meanings given in 
the preceding article. 

Taking moments around A, we have, 

Wg + fK-^elfi . t cos a = 

O' K^eli^ (ch sec a t sin a). 

We find also, t =: h (tan 00 — tan a) 5 
and since the moment Wg is the sum of the 
moments of the triangular prism AJDI^ and 


153 


the rectangular prism IDCEj minus the 
moment of the triangular prism BCE, all 
of the same density w, we readily find it 
to equal. 

— tan tan ^ (tan^ gd — tan^ /?) 

11 

~ ^ tan a li (tan go — — tan a) uv) 
or, 

uohj^ 

— (3 tan^u?—3 tan 09 tan ad-tanka'—tan^^)^ 


Observing that m—K^c sec a, on substi¬ 
tuting these values, and resolving with 
respect to tan go, we find. 


tan^ 09 -f- 

tan 09 I 2 ~Kx if cos a—<j sin a )—tan a\ 
, w A 

= 2 — I O' m-\-K^ tan a if cos oc —o' sin a) I 

wv. A 

tan^ a — tan^ f 


It will be observed that c does not ap¬ 
pear in this formula. This formula equally 
applies when the inner face of the wall 



154 


leans away from the earth, or B falls to the 
right of E, on simply replacing sin a and 
tan a by (— sin a) and (— tan a) through¬ 
out. As this formula is independent of h, 
it is true for all values of h. 

When h is given, tan gd is found from 
this formula, whence, t=h (tan gj — tan a). 

When a—oov the back of the wall is ver¬ 
tical, tan^u^-f-2 — 7fi/tan gd=2- () ^ 

tv tv 

inn^/3. 

For a rectangular wall, tan /?=oin last 
formula. 

69. The Practical Designing of Retaining- 
Walls .—By the use principally of the above 
formula, the quantities in the following 
table were made out by Leygue, using the 
value of m and given in Arts. 57 and 
60 from his experiments, and assuming 
the values 

— = i and f = tan cf> = 
w o 3 

The value a in columns 5, 7, and 6 gives 
the ratio of thickness t at base to the 
height h for 0 '=1, (T=3, and the centre of 


155 


pressure on the base at the outer middle 
third limit (column (7i) corresponding to 
no stress at i?; and the 11th column gives 
the correspondiug factor of safety for the 
latter case, which, it may be observed, lies 
between 1.19 and 3.57, and thus varies 
between such wide limits as to prove that 
the middle third limit is not so good a test 
of the stability of the wall as the use of a 
constant factor of safety o'. 

While, therefore, it is recommended that 
this limit be not exceeded, it is likewise 
recommended that the factor of safety o' 
be not taken below say 2.5, which about 
agrees with practice for rectangular walls 
(type 4). 


# 


Table No. 


156 


b 

^ in 

O 

00 QQ N 

Tf eo eo 

oi oi oi 

3.67 

3.19 

2.95 

[ 

1.90 

1.91 

1.94 

Surface, S. 

eo 

II s 

b 

0,271 

0.338 

0.392 

TOO I 

0.357 

0.427 

0.482 

1.266 

0.260 

0.327 

0.402 

0.989 

o - 

0.234 

0.288 

0.329 

0.851 

0 394 

0.442 

0.480 

1.316 

0.188 

0 245 

0 296 

0.729 

7 00 

b 

0.142 

0.161 

0.176 

0.479 

0.178 

0.214 

0.241 

CO 

C*^ 

O 

0.112 

0.147 

0.177 

0.436 

s' 

'S.i*' 

eo 

II t- 

b 

0.471 

0.538 

0.592 

l> v- c? 

1*^ 00 

TT iO lO 

o o o* 

0.360 

0.437 

0.502 


0.434 

0.488 

0.529 

0.494 

0.542 

0.580 

0.288 

0.345 

0.396 

1 « 
b 

I 1 1 

0 278 
0.314 
0.341 

0.212 

0.247 

0.277 

tan i 

4 

»-|iWM|o0 

O .-'|«'N|ffS 

O 

PI «> 
c3 

1 

1 

o 

o3 

-4^ 

o 

^111 

II vi ? 

o 

11 

13 

ci 

A V 

a 

ci 

4^ 

Type 

No. 

1 



CO 


































































Table No. 9.— Continued. 


157 


b 

O ^ 

2.51 

2.45 

2.40 

1.33 

1.41 

1.46 

1.14 

1.19 

1.23 

Surface, S. 

CO 

0.343 

0.424 

0.490 

1.257 

0.223 

0.189 

0.349 

0.761 

0.155 

0.219 

0.269 

CO 

CD 

o' 


5^ CO th 
T-H t-. CO 
CO CO 

o o o 

1.119 

0.121 

0.166 

0.208 

0.495 

0.066 

0.100 

0.129 

0.295 

ll 00 

b 

0.184 

0.224 

0.258 

0.666 

0.093 

0.124 

0.154 

0.371 

0,056 

0.086 

0.106 

0.248 

*8 

CO 

II t- 

b 

0.343 

0.424 

0.490 

0.223 

0.289 

0.349 

0.155 

0.219 

0.269 

o 

0.312 

0.376 

0.431 

0.121 

0.166 

0.208 

0.066 

0.100 

0.129 

ll in 

b 

0.184 

0.224 

0.258 

0.093 

0.124 

0.154 

0.056 

0.086 

0.106 

tan i 

4 

^ >-i|®ie«lo5 

O TH|oiOTleo 

O 

fl CO 
cS 

O 

th|io 

+ 

If 

d ^ 
cS 

43 

o 

11 

d 

n 


Type 

No. 

1 



CD 


































































158 


The columns 8, 9, and 10 give the areas 
o£ the cross sections o£ the walls £or a—l, 
(7=3, and the middle third limit and 

it is by an attentive study o£ these areas 
that the most economical type o£ wall £or 
any case can be £ound. For o' lying be¬ 
tween 2 and 3, the types are most econ¬ 
omical in the £ollowing order:— 

Nos. 6 and 5, leaning walls having parallel 
£aces ; 

No* 3, with the exterior £ace battered, and 
the inner £ace vertical j 
Noo 1, both £aces battered; 

No. 4, both £aces vertical; 

and lastly No. 2, with the interior £ace 

battered, and the exterior £ace vertical. 

Thus the leaning walls are decidedly 
the most economical; an(3, besides, the 
pressures are better distributed on the 
base. The £ormula does not apply to type 
No. 1 £or (7 = 1 as /? > 07, but the sur£aces 
£or (7 = 1 were made out £or a triangular 
type, having an interior batter o£ -i, and 
the angle /? as £ound by computation. 

70. For walls with projections at inter- 


159 


vals on the exterior or Interior, or for but¬ 
tressed or counterforted walls, theory does 
not show any economy over the leaning 
walls (types 5 and 6). In fact, for face- 
walls at the limit of stability, except 
for the buttresses, and for a factor of 
safety of the entire wall including but¬ 
tresses, about a line passing through the 


Fia. 13. 



outer toes of the buttresses of from 2 to 3, 
the surfaces S were about equal to those 
for type 5, corresponding to a mean of 
columns 8 and 10. Fig. 13 shows a good 
form of buttressed wall, with the face in the 
form of arches convex from the earth side ; 
but their hideousness militates against 
them. For counterforted walls, the lateral 
pressure of the earth against the sides 













160 


causes friction; and if (see Art. 56) we 
admit that the thrust on the face-wall is 
diminished and that the excess is sus¬ 
tained through the friction by the counter¬ 
forts. a computation shows that still the 
counterforted walls are no more economi¬ 
cal than type 3 with the back vertical and 
an exterior batter. It is likely that the 
thrust is less than assumed, as experiment 
has demonstrated the very great economy 
of counterforted walls. But it is a fact 
that in numerous cases the counterforts 
have been broken off from the face-wall, 
possibly from being carried down, through 
a bad foundation, by the settling of the 
earth, as well as from the earth-thrust; 
and they are not to be recommended, 
except as a support to leaning walls, during 
construction especially. To push economy 
to an extreme, a thin, leaning face-wall or 
mask is used, with counterforts at inter¬ 
vals, whose back faces are plumb; the 
space between being filled with earth well 
compacted, or with rough stones laid by 
hand. It is found that such a wall has 
nearly the same stability.as if the com- 


161 


pacted earth or rubble was a part o£ the 
wall (see ‘'Annales des Pouts et Chaus- 
sees’^ for November, 1885, and January, 
1887, for descriptions of such walls con¬ 
structed in France). Leygue actually 
constructed some small triangular coun¬ 
terforts in plaster, with an exterior slope 
of without any face-wall, which held 
the earth perfectly. The height was 1.31 
feet, the base 0.66 feet, and the thickness 
only 0.08 feet; the distances between the 
counterforts being from 0.33 to 0.49 feet, 
the intervals between being filled with plas¬ 
ter thrown in without especial precautions. 
The earth extended to the top of the 
counterforts, and with a surcharge besides. 

In the case of very high walls, it would 
prove economical to connect the counter¬ 
forts at various heights with arches (these 
could easily be constructed with an earth 
centring), which would immobilize a large 
portion of the earth filling, and tend to 
withdraw the active thrust farther to the 
rear. The space between the arches and 
counterforts should be entirely filled with 
compacted earth, to add to the stability. 


162 


7L We have now completed our task 
o£ giving complete theoretical and semi- 
empirical methods for the design of re- 
taining-walls, and have discussed all avail¬ 
able experiments by the different methods, 
and compared results, so that the engineer 
can better ap]3reciate their practical value 
in the practical designing of retaining- 
walls. We have found that although 
theory gave good results for vertical walls 
with earth level at the top, yet for many 
of the other cases it departed essentially 
from practice; so that for all cases the 
two semi-empirical methods are to be rec¬ 
ommended as giving the best results, sup¬ 
plemented when necessary for various 
values of <f> by the theoretical method. 

From preceding discussions, the engineer 
will doubtless draw some analogy between 
the theory of long columns and that o£ 
retaining-walls, from their inapplicability 
in certain limiting cases; and as in the 
case of columns the engineer has come 
finally to depend entirely on experiments^ 
the same may be confidently predicted of 
retaining-walls. 


163 


The experiments given in this treatise 
must he regarded, then, as only the first 
instalment on small-sized walls 5 and it is 
to be hoped that in the future the experi¬ 
ments will be extended to larger walls, and 
higher surcharges, and that by independ¬ 
ent methods of observation the amount of 
the earth-thrust, as well as its points of 
application, will be more definitely ascer¬ 
tained. 



4 


164 


APPENDIX I. 


DESIGN FOK A VEEY HIGH MASONKY 

DAM. 

Engineers are by no means agreed upon the 
proper profile to give high-masonry dams; although 
the three conditions, that there shall he no tension 
at any horizontal joints safe unit stresses every¬ 
where^ and no 'possible sliding along any plane joints 
seem to be generally accepted as essential to a good 
design. 

The writer suggests one more condition, that 
the factors of safety against overturning about any 
joint on the outer face shall increase gradually as 
we proceed upwards from the base, to allow for the 
proportionately greater influence, on the higher 
joints, of the effects of wind and wave action, ice, 
floating bodies, dynamite, or other accidental 
forces. The exact amount of increase must be 
largely a matter of judgment; but, if the principle 
is accepted, it can only result in making stromger 
dams. 



The accompanying sketch of a dam 258 feet 
high to the surface of water (see also “Engineering 
News” for June 23, 1888) satisfies the four condi- 


,24 



tions named, and will be briefly described. The 
dam is of the same total height (265 feet) and 
volume (nearly) as the proposed Quaker-Bridge 
dam, and, for ease of comparison, is designed, as 



















166 


was that dam, for masonry weighing times as 
much as water. The dam is 24 feet wide at top, 
38 feet wide, 50 feet below the surface of water (7 
feet below the top), and 196.1 feet wide at the 
base. The up-stream face is vertical for the first 
57 feet from the top, and then batters at the rate 
of 30 feet in 200 to the base. The outer face slopes 
uniformly from the top to 50 feet below the water 
surface, and then slopes uniformly to the base. 

The curves of pressure, for reservoir full or 
empty (the lines connecting the centres of pressure 
on the different horizontol joints are here styled 
the curves of pressure), are found as hitherto ex¬ 
plained, and are seen to lie well within the middle 
third of the base, so that the horizontal joints under 
the static pressure are only subjected to compres¬ 
sion throughout their w'hole extent. Further, it 
was found by construction, that if a horizontal 
force be assumed as acting at the surface of water, 
of such intensity (29,375 pounds) as to cause the 
total resultant, on the joint 50 feet below the w'ater 
level, to cut the joint one-third of its width from 
the outer face; then if this same force, acting at 
the surface of water, is combined in turn with each 
of the other resultants on the lower horizontal 
joints, the new centres of pressure will still lie well 
within the middle third for the low'er joints. To 
secure uniformity of results for all the joints, the 
width at the 50 feet level should be increased, 
although it is now much greater than ordinarily 


167 


% 

constructed. If, however, the effects of earthquake 
vibrations are to be guarded against, we cannot re¬ 
place them by the action of a single force acting at 
the surface, so that the increased width of the 
upper joints must be largely a matter of judgment. ^ 

The numbers to the right of the figure, in the 
form of a fraction, give for the corresponding 
joints, for the upper numbers, the factor against 
overturning, or the factor by which it is necessary 
to multiply the static horizontal thrust of the water 
to cause the total resultant to pass through the 
outer edge of the joint considered; and for the 
lower numbers, the ratio of the weight of masonry 
above a joint to the static thrust of water against 
it;’ which is, in a certain sense, a factor of safety 
against sliding on a horizontal joint. These factors 
are seen to increase from the base upwards, so that 
the suggested fourth condition is satisfied. 

1 It is stated in Engineering News for June 30, 1888, on 
the authority of Mr. Thomas C. Keefer, President Ameri¬ 
can Society of Civil Engineers, that “an ice bridge of 
about 90 feet span, between two fixed abutments, ex¬ 
panded so from a rise of temperature, as to rise 3 feet in 
the centre.” If we regard the arch thus formed as free 
to turn at the abutments and at the crown, we easily find 
for ice one foot thick, the horizontal thrust H exerted at 

62 5 

the abutments, from the equation, 3H= —u X ^ 52 , to be 

2 

in pounds per square foot 21,094 pounds. Much 
higher pressures may possibly be experienced sometimes 
near the top of high dams in northern latitudes, and it 
seems only proper to include such contingencies in their 
design. 



168 


The unit stresses, in pounds per square foot, at 
the outer edges of the joints for reservoir full, and 
at the inner edges for reservoir empty, are given 
in columns 4 and 5 of the following table, being 
computed from the formula 
6 w\ W 

T r 


p=/4 


-of Chap. I. 


Depth of 
Joint below 
Water 
Level. 

1 . 

Water 

Pressure. 

2 . 

W eights 
of 

Masonry. 

3. 

Pressure 

at 

outer edge. 

4. 

Pressure 

at 

inner edge. 

5. 

feet. 



lbs. 

lbs. 

50 

1,250 

4,417 

8,860 

10,460 

100 

5,000 

11,540 

13,480 

16,130 

150 

11,250 

23,420 

20,410 

21,440 

200 

20,000 

40,040 

27,330 

27,170 

250 

31,250 

61,420 

34,350 

33,130 

258 

33,282 

65,270 

35,360 

34,120 


The numbers of columns 2 and 3 for one foot in 
length of the wall are expressed in weights of cubic 
feet of water, and must be multiplied by 62.5 to 
reduce to pounds. 

The unit pressures, although necessarily high, 
are still permissible. By spreading the lower part 
of the dam still more, these unit stresses would be 
theoretically diminished, though it is likely that in 
reality the pressures at the positions of the old toes 
would not be very materially altered; but the 
masonry being surrounded with other masonry 
could, most probably, stand a higher pressure. 












169 


The unit pressures p given in columns 4 and 5 
are not the maximum normal pressures at the 
faces. In Appendix III (e), it is proved that the 
maximum normal stress at a face acts parallel to 
that face on a plane at right angles to it and that 
its intensity is given by the formula, f==psec^(f>, 
where 4> is the angle the face makes with the 
vertical. In this example, where ^ = 31° 23' for 
the outer face and 8° 32' for the inner face, the 


D 



Fig. 15. 


values of / at the outer and inner faces are found 
by multiplying the numliers given in columns 4 
and 5 by 1.37 and 1.02 respectively. 

The first derivation of the important formula, 
/=/> sec2^, has been credited to Levy by Dr. 
Unwin,! who likewise states that in several old 
dams which have lasted for centuries, the values 
of p, ranged from 11^ to 14 tons per square foot, 
giving the maximum compressive stresses / from 
15 to 20 tons per square foot (234 to 311 lbs. per 
square inch). 

! Minutes of Proceedings, Inst. C. E., Vol. CLXXII, 
Part II, p. 134. 





170 


The so-called factors against overturning are not 
true ones, for a computation shows that if the 
water pressure down to the joints 50, 100, and 150 
feet below the surface should become 2, 1|, 1^ 
times the original, respectively, that tension would 
just begin to be exerted at the inner face. This 
would happen for lower joints for thrusts about 1^ 
to 11 times the original. If, from any cause, as 
accidental forces at the top, earthquakes, etc., the 
thrusts should be increased over these amounts, 
causing tension at the inner edges beyond the 
capacity of the mortar to withstand, the joints 
would crack and open, water would get in, dimin¬ 
ishing the weight of the masonry materially, the 
centres of pressure would move outwards, and the 
unit pressures at the outer toes would very much 
increase, leading perhaps ultimately to the destruc¬ 
tion of the dam through sliding, overturning, or 
crushing at the down-stream face. 

We shall now consider the capacity of resistance 
of the dam to sliding along any oblique joint as 
AK. ^ Let AB represent, in magnitude and direc¬ 
tion, the resultant of the water pressure and weight 
of masonry on the horizontal joint AH, and let the 
vertical AB represent the w'eight of the triangular 
mass AHK, all for one foot in length of the wall. 
Draw BN j. AK and BN H to intersection N; 


1 See Annales des Fonts et Chauss^es for May, 1887. 



171 


then DN — component of BD normal to plane 
AK, and BN X tan Twhere tan </> = co-efficient 
of friction of masonry on masonry) is the total 
friction that can be exerted by the plane AN. If 
we lay off angle JVBE = 4> (taken as 35° here) to 
intersection £J with the parallel component BJY, 
we have BN tan — EN, so that BE must be 
resisted by cohesion; and the unit-shearing stress 

BE 

along the plane AK = ~ - . If, now, we produce 

AE 

KE on to intersection C, wuth AB produced, we 

BO 

have the unit shear represented by BN, which is a 

A C 

maximum, for various planes passing through A, 
tvhen G is farthest removed from B. 

On effecting this construction, then, for a series 
of planes passing through A, we quickly find the 
plane which will have to supply the maximum in¬ 
tensity of shear, or the plane of rupture, to lie near 
A K (there is very little difference for a series of 
planes lying near each other); and the shear per 
square foot required to resist sliding, in addition to 
the frictional resistance, to be about twenty-seven 
hundred and fifty pounds. To offer the greatest 
resistance to sliding, there should be no regular 
courses, and the stones should break joint verti¬ 
cally as well as horizontally, or the courses near the 
outer face should be curved so as to be approxi¬ 
mately normal to that face. For a retaining-wall 
of dry rubble, carelessly laid, we see that there is 
every probability of failure by sliding along some 
inclined plane. Here the stones must be carefully 


# 


172 


interlockf d to prevent sliding. For the reservoir- 
wall, where the best cement is used, and the joints 
are broken, there should be no fear of sliding when 
sufficient thickness is given to avoid tension. In 
the Habra dam, a hundred and sixteen feet high, 
this was not done; and the dam broke along a 
plane, passing through the outer toe nearly, and 
making the angle of friction of masonry on 
masonry with the horizontal. 

It is well to note, too, that friction alone will not 
prevent sliding along planes inclined not far from 
the horizontal as well as those below, so that a 
proper resistance to shear must be provided for in 
every dam. Possibly the weak point of many dams 
is in this very particular. 

The capacity of the dam in question to resist 
rotation about the toe of an inclined base may next 
be tried, and it will be formd to be stable; for the 
weight of masonry, as well as its arm, increases to 
counterbalance the increase of arm of the water- 
thrust. The dam thus satisfies all the conditions 
of stability; and, although some of its dimensions 
may be changed with advantage perhaps, it yet 
suffices very well to point out the principles of 
design. 

See Engineering News for January 12, 1893 and 
May 9, 1907 for effects of expansion of ice. 


% 


173 


APPENDIX II. 


STRESSES IN MASONRY DAMS.» 

The object of this investigation is to deter¬ 
mine the amounts and distribution of the stresses 
in a masonry dam, at points not too near the 
foundations, having assumed the usual “ law of 
the trapezoid,” that vertical unit pressures on 
horizontal planes vary uniformly from face to 
face. 

Experiment indicates that such vertical stresses 
increase pretty regularly in going from the inner 
to the outer face, for reservoir full, until we near 
the down-stream or outer face, where the stress 
gradually changes to a decreasing one, which 
decrease continues to the end of the horizontal 
section. The law of the trapezoid is thus only 
approximately true over part of the section, but, 
as it gives an excess pressure where it attains a 
maximum, it errs on the safe side. 


1 What follows in Appendices II and III was first 
given by the author in Trans. Am. Soc. C.E., Vol. 
LXIV, p. 208. 




174 


The profile of the dam selected is of the trian¬ 
gular type, with some additions at the top, but the 
method used in determining the stresses is general 
and will apply to any type of profile. The final 
equations will give, at any (interior or exterior) 
point of the horizontal section considerod, the 
vertical unit stress on the horizontal section, the 
normal stress on a vertical plane, and the unit 
shear on either horizontal or vertical planes. 
From these stresses, the maximum and minimum 
normal stresses, and the planes on which they act, 
can be determined, and ultimately, if desired, the 
stress on any assumed plane can be ascertained. 

The solution presented is approximate, which is 
justifiable, in view of the approximation involved 
in “the law of the trapezoid” used. The results, 
however, are practically correct, as will be evident 
from the checks applied, resulting from the exact 
theory given in Appendix III. The theory used, 
being simple, should be easily followed. 

Let Fig. 16 represent a slice of the dam con¬ 
tained between two vertical parallel planes, 1 ft. 
apart and perpendicular to the faces. The batter 


e . 130 0.65 

of OB IS-=-; 

200 1 


that of OE being 


4 _ 0.02 
200 


The batter of the inner face was found by trial, 
so that the centers of pressure on horizontal 
sections, for reservoir empty, should nowhere pass 
more than a fraction of a foot outside the middle 
third of the section. The simple type of profile 
shown was adopted for ease of computation. 

For convenience in subsequent computations, 
the breadths, b = EB, of horizontal sections, corre- 





175 


^ponding to various depths, h, below the surface 
of the water in the reservoir, are given, all dimen¬ 
sions being in feet: 


A = 199.0, 6 = 133.330 

6 = 199.5, 6 = 133.665 

6 = 200.0, 6 = 134.000 

6 = 200.5, 6 = 134.335 

6 = 201.0, 6 = 134.670. 


Take the weight of 1 cu. ft. of masonry equal 
to 1; then the weight of masonry above any 





Fig. 16. 


section is equal to the corresponding area in 
Fig. 16 above that section. The area of the por¬ 
tion above EOB is readily found to be 712, and 
its moment about the vertical, AO, is 11,603, the 
unit of length being the foot. In Fig. 16, D is 















176 


where the vertical through the center of gravity 
of the dam above the joint, EB, cuts that joint, 
and C is the center of pressure on that joint when 
the water pressure on EO is combined with the 
weight of masonry, W, above EB. 

As h varies, suppose each horizontal joint, in 
turn, marked similarly to the joint at h = 200, with 
the letters E, A, D', C, B; then, for any joint, on 
taking moments of the triangles, AOB, AOE, and 
the area above OB about A, we find 


AD 


A n 

— {AB^-EA^) +11,603 
6 

W 


Assuming that the masonry weighs 2^ times the 
water per cubic unit, then the weight of a cubic 
2 

foot of water is —. It would entail but little extra 
5 


trouble here, where the inner face has a uniform 
batter throughout, to include the vertical com¬ 
ponent of the water pressure on the face, EO\ 
but it will be neglected, as usual. 

The horizontal water pressure for the height, h, 


is thus, ^X —= — 
5 2 5 

5 3 15 


and its moment about C is 


Taking moments of W and water pressure about 
C, we have at once. 


dc=2-x 

15 


w‘ 



177 


From the last two formulas, we derive the 
following results: 


h W 

199 13978.335 

200 14112.000 

201 14246.335 


AD DC 

40.49141 37.58483 

40.70316 37.79289 

40.91488 38.00089 


A seven-place logarithmic table was used 
throughout, the aim in the computations being 
to get the seventh significant figure correct within 
one or two units. The necessity for this accuracy 
will be seen later. 

The distances EC and CB are now readily 
derived. 


For;i = 199, F;C=82.05624, (75 = 51.27376; 
/t=200, 5(7=82.49605, C5 = 51.50395; 
h=20\, 5C = 82.93577, 55 = 51.73423. 

On any plane, EB, the vertical unit pressure 


at 5=pi = ^-^_^lF, 
62 


^ „ 46 - 655 ^. 

at E=P 2 = - W; 

62 


where 6=55, and W is the weight of masonry 
above the plane. This follows from the assumed 
“law of the trapezoid.” 

From these formulas we derive: 

At 6 = 199, pi = 177.45483, p2 =32.22542 ; 

6 = 200, pi = 178.3855, p2 =32.24139; 
6=201, pi = 179.3160, p2 =32.25798. 




Call p the vertical unit stress at a distance, x', 
from E; then 



and the total str*«s on the base, x', is 



7^i-7>V 2 

2b 


. ( 1 ) 


To find the unit shear on vertical or horizontal 
planes,! consider a slice of the dam, bounded by 


! The writer desires here to acknowledge his indebted¬ 
ness to a recent paper on “Stresses in Masonry Dams,’’ 
by Ernest Prescot Hill, M. Inst. C.E., published in 
Minutes of Proceedings, Inst. C.E., Vol. CLXXII, 
p. 134. Mr. Hill considers the case of a dam with a 
vertical inner face. By the aid of the calculus, he 
effects an exact solution, which leads to general formulas 
for shear and normal pressures on vertical planes. 

The principles at the base of his method, though 
somewhat disguised by the calculus notation, are essen¬ 
tially the same as those used by the author. 

Mr. Hill ascribes to Professor W. C. Unwin the sugges¬ 
tion, “that the shearing stress at any point may be 
found by considering the difference between the total 
net vertical reactions [between that point and either 
face] along two horizontal planes at unit distance 
apart,” and states that Prof. Unwin, “has applied 
the principle to a triangular dam by the use of alge¬ 
braical methods.” 

Dr. Unwin states (Proc. Inst. C.E., Vol. CLXXII, 
Part II, p. 161) that he ascertained after his papers 
were written, that by a different method, Levy had 
previously arrived at the same conclusions. 





179 


horizontal planes at ;i = 199 and A = 200, the water 
_f&CB and a vertical plane, at a distance, x, from the 
inner face (Fig. 17), in equilibrium under the water 
pressure acting horizontally on its left face and 
the forces exerted by the other parts of the dam 
on the slice. These forces consist of the uniformly 
increasing stress, P', on top, acting down; the 
uniformly increasing stress, P, on the bottom, 
acting up; a shear acting on the vertical plane 


A=199 
A«=300 E 



X 1 


t I j t f I I .J 


Fig. 17, 


at the right, of average intensity gi per square 
foot, the weight of the body (x —0.01), besides the 
horizontal forces to be given later. The vertical 
component of the water pressure is here neglected, 
as usual. The origin for x is taken, here and in 
all subsequent work, at the level, A = 200, at the 
inner face. 

For equilibrium, the sum of the vertical com¬ 
ponents must be zero. 

Therefore, 

9i = (x-0.01)+P'-P. ... (2) 























ISO 


To find P', substitute in Equation (1), 

— 0.02, P2=32.22542, pi —p2 = 145.22941, k = 
133.330, giving P'=32.20364a:+0.5446238x*- 
0.6442906. For P, x' =x, p2 = 32.24139, pi —p2=* 
146.1441, and 6 = 134; therefore, 

P =32.24139X + 0.5453138x2. 

Substituting in Equation (2), we derive the 
average unit shear, 

gi=—0.6542906 —0.96225X —0.0006900x2. . ( 3 ) 

This value of q\ is strictly correct w'.en x^0.02. 
It is slightly in error when 0<x<0.02. 



Fig. 18 . 


A similar investigation holds to obtain the 
average unit shear, 52 (Fig. IS), on a vertical plane, 
at a distance, x, from E, extending from the 
level, 6. = 200, to the level, /i = 201. 

We have, for equilibrium, 

. ^ ^ ^ _g2 =^(x+ 0 . 01 )+P-P". 


( 4 ) 


























181 


We find P" by substituting in Equation ( 1 ), 
x' = (a: + 0.02), p 2 =32.25798, pi-p 2 = 147.05802, 
and 6 = 134.67. P" =32.27982rc+ 0.5459941a:2+ 
0.6453780. Substituting this, and the value pre¬ 
viously found for P, in Equation (4), we derive, 

52 = -0.6353780+ 0.90157a:-0.0006803x2. (5) 

This is strictly correct only when x> 0 . 

The mean, ^( 51 + 52 ), of these average shears will 
be assumed approximately equal to the inten¬ 
sity of shear at the point, G{x = EG), at the level, 
h = 200. Call q this intensity of shear on a ver¬ 
tical plane at G) therefore, 

5 = -0.6448343 +0.96191x-0.0006856x2. ( 6 ) 

Checks .—By Appendix HI ( 6 ) and (d), the exact 
value of 5 ', at either face, =p tan where p = 
vertical unit normal stress at the face and 4> is 
the angle the face makes with the vertical. Thus, 
at the inner face, 5 = —32.24139X0.02 = —0.6448278, 
whereas Equation ( 6 ) gives for x = 0, 5 = — 

0.6448343. 

At the outer face, the exact value is, 178.3855 
X0.65 = 115.9506, whereas Equation ( 6 ) gives, for 
x = 134, 5 = 115.9405. 

A still more searching test can be devised. It 
is a well-known principle that the intensity of 
shear at a point, on vertical or horizontal planes, 
is the same [Appendix III (a)]. Therefore, regard¬ 
ing Equation ( 6 ) as giving the horizontal unit 


182 


shear, at the level, ^ = 200, where 6 = 134 ft.; the 
total shear, from face to face, on this level, is 


x=134 

= 7999.75. 

= 0 



This should equal the total water pressure down 



thus gives practically exact results. 

In order to find the normal unit stress on a 
vertical plane, we shall assume that q\, given by 
Equation (3), equals the intensity of shear on a 
vertical or horizontal plane at the point, x, at 
6 = 199.5; and that 52 , given by Equation (5), 
gives the shear intensity at x at 6 = 200.5. This 
evidently supposes that the shear intensity in¬ 
creases uniformly, vertically, from 6 = 199 to 
6 = 201 . 

Consider a portion of the dam, Fig. 19, bounded 
by the water face; the plane, FM, 4t the level, 
6 = 199.5, on which the total shear is Q', the plane 
EN, at the level 200-5, on which the total shear 
is Q, and the vertical plane, MN, 1 sq. ft. in area, 
on which the average normal stress is p'. The 
water pressure on EF will be supposed to be 
exerted horizontally. It is equal to 80 units. 
Assuming, as stated, that gi = intensity of hori¬ 
zontal shear at M, and 32 = the corresponding 
intensity at N, we have, taking the origin as 
before at 0 , 


Q' = [ dx] Q = Ix] 




183 


or, 

0' = 0.006494794 - 0.6542906x + 0.481125^2 

-0.00023x»; 

Q=- 0.00640186 - 0.6353780x + 0.480785x2 

X* 

-0.0006803-3. 


Checks '.—The total water pressure for A = 199.5 

is-^(199.5)2 = 7960.05 and for A = 200.5, ^(200.5)2 

= 8040.05. The first should equal Q', for x = 
133.665, or 7959.22; the second should equal Q, 


Q' 



X= 0 


x—O.Ol 

X 


G 


X 0.01 






p* X 1 


Q 


Fig. 19. 

for x = 134.335, or 8041.12. The slight differences 
tend to give confidence in the results. 

For equilibrium, the sum of the horizontal forces 
acting on EFMN, Fig. 19,must be zero; therefore, 

p'=80+Q — Q , .... (7) 

p'=80.01-0.0189X +0.00034x2-0.00000323x2. 

This average stress will now be assumed to be the 
intensity of the horizontal unit stress on vertical 
planes at A = 200. 











184 


It will now be perceived why a seven-place table 
was necessary in the computations, the coefficients 
of and having only two or three significant 
figures in the final result. If the planes originally 
had been taken 0.1 ft. apart vertically, a ten-place 
table would have been required. 

Checks. —The value of p', for x=0, p' =80.012896, 
is the same as that given by Appendix III (d), 
80+0.6448X0.02. When a: = 134, the formula 
gives p' = 75.81, whereas the exact theory. Appen¬ 
dix III (6), gives p'=m2p = (0.65)2X 178.39 = 75.37. 
The difference is 0.44 at the outer face. For any 
other point, it might be assumed to vary with x, 
so that it could be corrected by substracting 
0.44 

yg;^a; = 0.0033x from the value of p' above. For 
ease of computation, the formula will be written, 

p' =80.01 - 0.02a: + 0.00034a:2 - 0.0000l|-’. (8) 

The first coefficient of x^ cannot be counted on to 
the last two figures, hence we are permitted to 
change 323 to 333 in that coefficient. When 
x = 134. Equation (8) gives p'=75.41, nearly the 
exact value. 

The three formulas for p, q, and p', at the 
level h = 200, are thus as follows: 

p =32.24+ 1.0906.3x; 

5 = - 0.64 +0.962a: - 0.000686a:2; 

p' = 80.01 - 0.02a: + 0.00034a:2 - O.OOOOl^^. 

o 



185 


Since the weight per cubic foot of masonry was 
assumed as two and one-half times that of water, 
we must multiply the stresses given in Table I 
5 

t>y *2 = 156.25, to reduce to pounds per square 

foot; or by 1.085, to reduce to pounds per square 
inch. 

TABLE I. 


X 

0 

10 

25 

50 

V . 

32.24 

43.15 

59.50 

86.77 

Q . 

-0.64 

8.91 

22.98 

45.75 

P' . 

80.01 

80.02 

79.66 

79.11 

Max. /. 

80.02 

82.06 

94.67 

128.85 

Min. /. 

32.23 

41.11 

44.48 

37.03 

6 for max. /. . 

90° 46' 

77° 06' 

1 ^ 

1 Oi 

1 o 

o 

42° 36 


X 

75 

100 

134 

P . 

114.04 

141 .30 

178.39 

q . 

67 .65 

88.70 

115.95 

p'. 

79.01 

78.08 

75.37 

Max. /. 

166.40 

203.85 

253.71 

Min. /. 

26.64 

15.52 

0 

6 for max. /. . 

37° 44' 

35° 12' 

33° 01' 


In Table 1 the stresses are those experienced at 
the level, /?. = 200. 

p= vertical unit stress on a horizontal plane; 
g = shearing unit stress on horizontal or ver¬ 
tical planes; 

p'=horizontal unit stress on vertical planes; 
Max./= maximum normal stress acting on a plane 
inclined to the horizontal at the angle, 
0, given on the last line; 
































186 


Min./= minimum normal stress acting on a plane 
perpendicular to the last. 

From max. / and min. f, with 8, the ellipse of 
stress can be drawn, and the stress in any direc¬ 
tion, with the plane on which it acts, can be 
ascertained. 

It will be observed that there is no tension 
exerted anywhere, and that the maximum com¬ 
pression is 253.71, or 275 lbs. per square inch, 
which is exerted at the outer face, parallel to that 
face, upon a plane at right angles to the face. 

In Appendix III (e), the important formula, for 
the maximum normal intensity at the outer face, 
acting parallel to that face. 


cos^^’ 

is proved. In this instance, p = 178.39, tan ^== 
0.65, therefore ^=33° 01', whence /=253.71. 

This stress is unaccompanied with any conju¬ 
gate stress, perpendicular to the face. In the 
interior of the dam, where conjugate stresses 
prevail, the masonry is perhaps better able to 
withstand a certain compressive stress than at the 
face. The distribution of stresses, at the level, 
A = 200, is shown in Fig. 20, on the supposition 
that the base of the dam is a little below that 
level. The connection with the foundation mate¬ 
rially modifies this distribution; but Fig. 20 shows 
the distribution for sections, say, from 10 to 20 ft. 
above the base, up to the level h = 100, fairly well, 
on the basis of the trapezoid law. As has been 



187 


mentioned before, this law gives a pressure greater 
than the actual at the outer face. 

Since the batter of the inner face is very small, 
the results of Table I should agree approximately, 
except near the inner face, with those found by 
Mr. Hill in the paper referred to in the foot note. 


«-0 10 as so 7S lOO IM 



Substituting numerical values, Mr. Hill’s formulas, 
for h = 200, reduce to 

o = 0.9426X - 0.0005768x2, 

p'=80-0.0001289x2-0.0000009615x2; 


giving: 






















188 


X 

0 

10 

25 

50 

75 

100 

134 

Q 

0 

9.36 

23.20 

45.69 

67.45 

88.49 

115.95 

V' 

80 

79.99 

79.90 

79.56 

78.87 

77.75 

75.38 


On comparnig these formulas with those of the 
writer, it will be observed that the absolute term 
in the value of q and a consequent term of the 
first degree in x, in the value of p', are lacking 
in Mr. Hill’s formulas. This results from taking 
the inner face as vertical. Although the coeffi¬ 
cients also differ, it is seen that the numerical 
values are very nearly the same. 

In Fig. 21 are shown, on a drawing of the dam, 
to scale, the lines of the centers of pressure for 
reservoir full and empty. 

To the right, and under the word “factors,” 
are certain numbers, written in the form of frac¬ 
tions. For any joint, the upper number gives 
the factor against overturning, or the number by 
which it is necessary to multiply the water pressure 
down to the joint, to cause the total resultant to 
pass through the outer edge of the joint con¬ 
sidered. The lower numbers give the ratio of the 
weight of masonry above a joint to the water 
pressure corresponding. 

It is believed that these “factors” should in¬ 
crease from the base upward, to allow somewhat 
for earth(!juakes, expansion of ice in freezing, etc., 
since the effects of such accidental forces is pro¬ 
portionately greater on the upper joints. 

Stresses due to water infiltration are not included 

















189 


here; neither are stresses due to temperature 
changes. 

The unit stresses, /, in pounds per square inch, 
acting parallel to the adjacent face, are as follows, 


so 



and refer to the outer edges of the joints, for 
reservoir full, and to the inner edges for reservoir 


empty: 

h 

50 

100 

150 

200 


/ at Outer Edge, 

85 

136 

204 

275 


at Inner Edge. 

58 

133 

180 

228 


















190 


The stresses, /, are normal pressures on planes 
perpendicular to the respective faces, and are the 
greatest stresses that can be experienced in the 
dam. In fact, they are greater than the true 
stresses, since the trapezoid law is not exact, 
particularly near the base, as before remarked. 

It would then seem that the dam, thus far, is safe, 
since the maximum unit stress is less than con¬ 
crete, even, is subjected to daily, in good practice. 

For an actual construction, the outer face should 
be curved, from near A = 50 to the top, as shown 
by the curved dotted line in Fig. 21. 

The subject of the stresses in masonry dams 
has caused a great deal of discussion among 
British engineers in the last two or three years. 
The subject was reopened by Mr. L. W. Atcherly » 
and Professor Karl Pearson, i who gave the results 
of certain experiments which seemed to indicate 
considerable tension across vertical planes near the 
outer toe. The late Sir Benjamin Baker, Hon. 

M. Am. Soc. C. E., also published ^ the results of 
experiments on a model dam of stiff jelly, and 
very recently, the “Experimental Investigations” 
of Sir J. W. Ottley and Mr. A. W. Brightmore * * on 
elastic dams of “plasticine” (a kind of modeling 
clay) and the experiments of Messrs. J. S. Wilson 
and W. Gore'* on “India Rubber Models” hav e 
been presente d. 

1 Minutes of Proceedings, Inst. C. E., Vol. CLXII, p. 
456. 

2 Ibid., Vol. CLXII, p. 123. 

* Ibid., Vol. CLXXII, p. 89. 

< Ibid., Vol. CLXXII, p. 107. 



191 


It is not the object of this paper to discuss these 
later experiments; but it may be remarked that 
they show very plainly that no tension exists near 
the outer toe, but that tension does exist at 
the inner toe, where the dam is joined to the 
foundation, and it has become a serious matter 
how to deal with it. The influence of the founda¬ 
tion in modifying the distribution of the stresses 
at the base of the dam was found to be very great, 
causing the shear there to be more uniform than 
higher up, where the parabolic law, nearh^ as given 
by the formulas above, was found to hold. Also, 
above some undertermined plane, a small distance 
above the base, the usual ‘'law of the trapezoid” 
was found to be approximately correct, leading to 
stresses on the safe side at the outer toe. This 
law leads to stresses at the outer toe of the base 
considerably in excess of the true ones. 

It was found, from the rubber models particu¬ 
larly, as theory indicates, that the greatest normal 
pressures are exerted at the down-stream face, 
for reservoir full, and they act in a direction 
parallel to that face. 


192 


APPENDIX III. 


RELATIONS BETWEEN STRESSES AT ANY 
POINT OF A DAM. 

(a) Consider a cube of masonry, Fig. 22, the 
edge of which has the length, a, bounded by ver¬ 
tical and horizontal 
planes and subjected to 
normal and shearing 
forces, caused by the 
action of the other parts 
of the dam. Since a 
will be supposed to di¬ 
minish indefinitely, the 
weight of the cube, 
which is proportional to 
a*, is an infinitesimal of 
the third order, and can 
be neglected in comparison with the normal forces, 
which vary as and are thus of the second order. 

Similarly, the average unit stresses exerted on 
the faces can be treated from the first as the unit 
stresses at any point. A, of the cube. As a 
diminishes indefinitely, the oppositely directed 



Fig. 22. 









193 


normal forces approach equality and balance 
independently; hence the couples formed by the 
shears on opposite faces must likewise approach 
equality; the one being right-handed, the other 
left-handed; therefore qaXa = q'aXa, or q=q'’, 
hence, the intensities of shear at a point on two 
planes at right angles are equal. The relative 
directions of the shears on two planes at right 
angles are determined, as above, from the con¬ 
sideration that one resulting couple must be right- 
handed and the other left-handed. This applies 
also to Figs. 23 to 26. 

JB 



(6) In Fig. 23, ABC is the right section of a 
prism at the outer face, with lateral faces one unit 
in length, perpendicular to the plane of the paper. 
Let AB be vertical; tan (f> = m, a constant; 

p = normal intensity on a horizontal plane at 

C; 

p' = normal intensity on a vertical plane at C; 
g = shear intensity on horizontal or vertical 
planes at C. 








194 


The weight of the prism is ^ab. 

Balancing vertical as well as horizontal com¬ 
ponents, we have, when a = AB and b=AC are 
very small, 

pb = qa + ^ab, nearly; 
p'a — qb. 

Dividing the first equation by b, the second by 
a, the limit, as a and b approach zero, gives exactly, 

p = q cot therefore q = mp; 

p'=q tan therefore p'=m^p, pp' = q^. 

These equations give the relations between p, 
q, and p' at the outer face. The same relations 
hold at the inner face, for reservoir empty, on 
replacing ^ by the angle the inner face makes 
with the vertical. 

For the remaining cases, the final limits will be 
written at once, since the complete process of 
deriving them is evident from the above. In fact, 
the weight of the prism, ^ab, being of the second 
order, can be neglected in comparison, with 
qa, etc. 

2 

(c) For reservoir full, calling w = -^h, the inten- 

o 

sity of water pressure, horizontally or vertically, 
at C, we have at the inner face, putting tan <}>' —n, 
Fig. 24, 

pb = qa + wb; p'a = qb +wa; 

1 

p=—q+w; p =qn + w. 

Tt 


therefore 


195 


(d) If the vertical component of the water 
pressure is neglected, these equations reduce to 


p'=qn + w, 


therefore 


q = pn; -p' -{-w. 



Fig. 24. 


Fig. 25. 


(e) Since the shear on the outer face is zero, 
therefore, by (a), the shear on a plane, AD, Fig. 25, 
perpendicular to the outer face, is al.so zero, or 
the stress on AD is normal. 

Call / the intensity of such a stress at C. The 
total pressure on AD=f'KAD=fb cos 4), and its 
vertical component is fh cos^ </>, therefore balancing 
the vertical components. 


pb=/b cos20; 


therefore 














196 


This is a most important formula for finding 
the maximum normal intensity at the outer face. 
It applies equally to the inner face for reservoir 
empty, on changing (j) to <j)', the angle the inner 
face makes with the vertical. For either face, p 
is the vertical normal unit stress at the face con¬ 
sidered. 

(/) Principal Normal Stresses at Any Point in 
the Dam and the Planes on which they Act. —In the 
prism, ABC, Fig, 26, let AB be one of the planes 



Fig. 26. 


on which the stress is normal. Let / be its inten¬ 
sity. The stress on the plane, AB, of unit length 
perpendicular to the plane of the paper, is thus 
/c; its vertical component is fc cos B =fb, and its 
horizontal component is fc sin B = fa, B being the 
angle that AB makes with the horizontal. 

Place the sum of the vertical forces acting on 
ABC equal to zero; also place the sum of hori¬ 
zontal forces equal to zero. 








197 


fh — pb-\-qa, therefore f—p =qt2in9,. 
fa = qh + p'a, therefore f—p'=q cot d. 

The difference of the last two equations gives 


, ^ ^ 1 — tan2 8 

p — p =g(cot ^-tan 8) 


2 tan 8 2q 

therefore tan 28 = z. --—7- =-;. 

1 —tan^(^ p — p 

\ 

The angles, 8 (differing by 90°), computed from 
this equation, give the directions of the planes, 
AB, on which the stress is entirely normal. 

From an equation above, we likewise have 


tan 8 = 


f-P 


This gives directly the plane on which a given 
/ acts. 

To deduce a formula for /, take the product of 
two equations above: 


(/-P)(/-P')=9^ 


therefore f =^[p+p' ± (p — 

This equation gives the two values of / corre¬ 
sponding to the two planes mentioned; com- 


\ 






108 


pressive when / is positive, tensile when negative. 
There can be no tension when pp' > g*. 

A better form for computation is, 

f=^\P + P' (V^^) *+4g»J. 


9 



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the; van nostrand science series. 


No. 45. THERMO-DYNAMICS. .New edition, in 

press. 

No. 4(i. ICE-MAKING MACHINES. From the 

French of M. Le Doux. Revised by Prof. 
J. E. Denton, D. S. Jacobus, and A. Riesen- 
berger. Sixth edition, revised. 

No. 47. LINKAGES: THE DIFFERENT FORMS 

and Uses of Articulated Links. By J. D. C. 
De Roos. 

No. 48. THEORY OF SOLID » AND BRACED 

Elastic Arches. By William Cain, C.E. 

No. 49. MOTION^ OF A SOLID IN A FLUID. By 

Thomas Craig, Ph.D. 

No. 50. DAVELLING-HOUSES: THEIR SANI- 

tary Construction and Arrangements. By 
Prof. W. H. Corfleld. 

No. 51. THE TELESCOPE; OPTICAL PRINCI- 

ples Involved in the Construction of Re¬ 
fracting and Reflecting Telescopes, with a 
new chapter on the Evolution of the Mod¬ 
ern Telescope, and a Bibliography to date. 
AVith diagrams and folding plates. By 
Thomas Nolan. Second edition, revised anU 
enlarged^ 

No. 52. IMAGINABLE QUANTITIES: THEIR GE- 

ometrical Interpretation. Translated from 
the French of M. Argand by Prof. A. S. 
Hardy. 

No. .53. INDUCTION COILS; HOW MADE AND 

How Used. Eleventh American edition. 

No. 54. KINEMATICS OF MACHINERY. By 
Prof. Alex. B. W. Kennedy. AVith an intro¬ 
duction by Prof. R. H. Thurston. 

No. 55. SEALER GASES: THEIR NATURE AND 

Origin. By A. de Varona. Second edition, 
revised and enlarged. 

No. 56. THE ACTUAL LATERAL PRESSURE 

of Earthwork. By Benj. Baker, M. Inst., 

C.E. 

No. 57. INCANDESCENT ELECTRIC LIGHTING. 

A Practical Description of the Edison Sys¬ 
tem. By L. H. Latimer. To which is added 
the Design and Operation of Incandescent 
Stations, by ,C. J. Field; and the Maximum 
Efficiency of Incandescent Lamps, by John 
W. Howell. 

No. 58. A^ENTILATION OF COAL MINES. By AAA 
Fairley, M.E., and Geo. J. Andre'. 










THU VAN NOSTRAND SCH^NCE SERIES. 


No. .50. RATIiROAD ECONOMICS; OR, NOTES 

■VYitli Comments. By S. W. Robinson, C.E. 

No. «0. STRENGTH OF WROUGHT-IRON 

Bridge Members. By S. W. Robinson, C.E. 

No. «1. POTABGE WATER, AND METHODS OP 

Detecting Impurities. By M. N. Baker. Sec¬ 
ond, ed., revised and enlarged. 

No. C2. THEORY OF THE GAS-ENGINE. By 

Dougald Clerk. Third edition. With addi¬ 
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No. 63. HOUSE-DRAINAGE AND SANITARY 
Plumbing. By W. P. Gerhard. Twelfth edi¬ 
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No. 64. ELECTRO-MAGNETS. By A. N. Mans¬ 
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No. 65. POCKET LOGARITHMS TO FOUR 

Places of Decimals. Including Logarithms 
of Numbers, etc. 

No. 66. DYNAMO-ELECTRIC MACHINERY, By 

S. P. Thompson. With an Introduction by 
• F. L. Pope. Third edition, revised. 

No. 67. HYDRAULIC TABLES FOR THE CAL- 

culation of the Discharge through Sewers, 
Pipes, and Conduits. Based on “Kutter’s 
Formula.” By P. J. Flynn. 

No. 68. STEAM-HEATING. By Robert Briggs. 

Third edition, revised, with additions by 
A. R. Wolff. 

No. 69. CHEMICAL PROBLEMS. By Prof. J. C. 

Foye. Fourth edition, revised and en¬ 
larged. 

No. 70. EXPLOSIVE MATERIALS. By Lieut. 

Jphn P. Wisser. 

No. 71. DYNAMIC ELECTRICITY. By John 

Hopkinson, J. N. Shoolbred, and R. E. Day. 

No. 72. TOPOGRAPHICAL SURVEYING. By 

George J. Specht, Prof. A. S. Hardy, John B. 
McMaster, and H. F. Walling. Third Edition, 
revised. 

No. 73. SYMBOLIC ALGEBRA; OR, THE ALGE- 

bra of Algebraic Numbers.. By Prof. Wil¬ 
liam Cain. 

No. 74. TESTING MACHINES: THEIR His¬ 
tory, Construction and Use. By Arthur V. 
Abbott. 




THE VAN NOSTRAND SCIENCE SERIES. 


x\o. 75. RECENT PROGRESS IIV DYXATIO- 

electric Machines. Being- a Supplement to 
“Dynamo-electric Machinery.’’ By Prof. 
Sylvanus P. Thompson. 

No. 7fi. MODERN REPRODUCTIVE GRAPHIC 

Processes. By Lieut. James S. Pettit,'U.S.A. 

No. 77. STADIA SURVEYING. The Theory of 

Stadia Measurements. By Arthur Winslow. 
Sixth edition. 

No. 78. THE STEAM-ENGINE INDICATOR 

and Its Use. By W. B. Le Van. 

No. 7». THE FIGURE OP THE EARTH. By 

Frank C. Roberts, C.E. 

No. 80. HEALTHY FOUNDATIONS FOR 

Houses. By Glenn Brown. 

No. 81. WATER METERS: COMPARATIVE 

Tests of Accuracy, Delivery, etc. Distinc¬ 
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Siemens, and Hesse meters. By Ross E. 
Browne. 

No. 82. THE PRESERVATION OF TIMBER BY 

the Use of Antiseptics. By Samuel Eagster 
Boulton, C.;^. 

No. 8,3. MECHANICAL INTEGRATORS. By Prof. 
Plenry S.' H. Shaw, C.E. 

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Prof. De Volson Wood. 

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termination. Description, and Classification 
of Minerals Found in the United States. By 
Prof. J. C. Foye. Fifth edition, revised. 

No. 87. TREATISE ON THE THEORY OF THE 

Construction of Helicoidal Oblique Arches. 
By John L. Culley, C.E. 

No. 88. BEAMS AND GIRDERS. Practical For¬ 
mulas for their Resistance. By P. H. Phil- 
brick. . 

No. SO. MODERN GUN COTTON: ITS MANU- 

facture, Properties, and Analyses. By Lieut. 
John P. Wisser, U.S.A. 

No. 00. ROTARY MOTION AS APPLIED TO 

the Gyroscope. By Major J. G. Barnard. 









THE VAN nostrand , SCIENCE SERIES. 


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metric, and Spirit. By Prof. I. O. Baker. 
Second edition. 

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Use. By Boverton Redwood, F.I.C., F.C.S. 

No. 93. RECENT PRACTICE IN THE SANI- 

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liam Paul Gerhard, C.E. 

No. 94. THE TREATMENT OP SEWAGE. By 

Dr. C. Meymott Tidy. 

No. 9.5. PLATE-GIRDER CONSTRUCTION. By 

Isami Hiroi, C.E. Fourth edition, revised. 

No. 96. ALTERNATE CURRENT MACHINERY. 
By Gisbet Kapp, Assoc. M. Inst., C.E. 

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Wastes. Second edition. By W. Paul Ger¬ 
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Amateurs. How to Wind for Any Output. 
By Frederick Walker. Fully illustrated. 
Third edition. 

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Edited with notes, etc., by F. E. Idell, M.E. 

No. 100. HOW TO BECOME AN ENGINEER; or, 

The Theoretical and Practical Training nec¬ 
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Engineer. By Prof. Geo. W. Plympton. 

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Mathematical Instruments. With Practical 
Hints for their Adjustment and Use. By F. 
R. Brainard, U. S. Navy. 

No. 102. THE GALVANIG CIRCUIT INVESTI- 

g ated Mathematically. By Dr. G. S. Ohm, 
erlin, 1827. Translated by William Fran¬ 
cis. With Preface and Notes by the Editor, 
Thomas D. Lockwood, M.I.E.E. Second edi¬ 
tion. 

No. 103. THE MICROSCOPICAL EXAMINATION 

of Potable Water. With Diagrams. By Geo. 
W. Rafter. Second edition. 

No. 104. VAN NOSTRAND’S TABLE-BOOK FOR 

Civil and Mechanical Engineers. Compiled 
by Prof. Geo. W. Plympton. 




THK VAN NOSTRAND SCIENCE SERIES. 


No. 105. DETERMINANTS. An Introduction to 

the Study of, with Examples and Applica¬ 
tions. By Prof. G. A. Miller. 

A"o. 106. COMPRESSED AIR. Experiments upon 

the Transmission of Power by Compressed 
Air in Paris. (Popp’s System.) By Prof. 
A. B. W. Kennedy. The Transmission and 
Distribution of Power from Central Stations 
by Compressed Air. By Prof. W. C. Unwin. 
Edited by F. E. Idell. Third edition. 

No. 107. A GRAPHICAL. METHOD FOR SWING 

Bridges. A Rational and Easy Graphical 
Analysis of the Stresses in Ordinary Swing 
Bridges. With an Introduction on the Gen¬ 
eral Theory of Graphical Statics, with Fold¬ 
ing Plates. Second edition. By Benjamin B'’, 
La Rue. 

No. 108. SLIDE-VALVE DIAGRAMS. A Preneh 

Method ' for Constructing Slide-valve Dia¬ 
grams. By Lloyd Bankson, B.S., Assistant 
Naval Constructor, U. S. Navy. 8 Folding 
Plates. 

No. 109. THE MEASUREMENT OP ELECTRIC 

Currents. Electrical Measuring Instruments. 
By James Swinburne. Meters for Electrical 
Energy. By C. H. Wordingham. Edited, 
with Preface, by T. Commerford Martin. 
With Folding Plate and Numerous Illustra¬ 
tions. 

No. 110. TRANSITION CURVES. A Field-book 

for Engineers, Containing Rules and Tables 
for Laying out Transition Curves. By Wal¬ 
ter G. B''ox, C.E. Second edition. 

No. 111. GAS-LIGHTING AND GAS-PITTING. 

Specifications and Rules for Gas-piping. 
Notes on the Advantages of Gas for Cook¬ 
ing and Heating, and Useful Hints to Gas 
Consumers. Third edition. By Wm. Paul 
Gerhard, C.E. 

No. 112. A PRIMER ON THE CALCULUS. By 

E. Sherman Gould, M. Am. Soc. C. E. Third 
edition, revised and enlarged. 

No. 113. PHYSICAL PROBLEMS and Their So¬ 
lution. By A. Bourgougnon, formerly As¬ 
sistant at Bellevue Hospital. Second ed. 

No. 114. USE OP THE SLIDE RULE. By 

F. A. Halsey, of the “American Machinist.’’ 
Fourth edition, revised and enlarged. 




THi; VAN NOSTRAND SCIENCE SERIES. 


No. 115. TRAVERSE TABLE. Showing the Dif¬ 
ference of Latitude and Departure for Dis¬ 
tances Between 1 and 100 and for Angles to 
Quarter Degrees Between 1 Degree and 90 
Degrees. (Reprinted from Scribner’s Pocket 
Table Book.) 

No. 116. WORM AND SPIRAL GEARING. Re¬ 
printed from “American Machinist.’’ By F. 
A. Halsey. Second revised and enlarged 
edition. 

No. 117. PRACTICAL HYDROSTATICS, AND 

Hydrostatic Formulas. With Numerous Il¬ 
lustrative Figures and Numerical Examples. 
By E. Sherman Gould. 

No. IIS. TREATMENT OF SEPTIC SEWAGE, 

with Diagrams and Figures. By Gleo. W. 
Rafter. ^ 

No. 119. LAY-OUT OF CORLISS VALVE GEARS. 

With Folding Plates and Diagrams. By 
Sanford A. Moss, M.S., Ph.D. Reprinted 
from “The American Machinist,” with revi¬ 
sions and additions. Second edition. 

No. 120. ART OF GENERATING GEAR TEETH. 

By Howard A. Coombs. With Figures, Dia¬ 
grams and Folding Plates. Reprinted from 
the “American Machinist.” 

No. 121. ELEMENTS OF GAS ENGINE DE- 

sign. Reprint of a Set of Notes accompany¬ 
ing a Course of Lectures delivered at Cor¬ 
nell University in 1902. By Sanford A. 
Moss. Illustrated. 

No. 122. SHAFT GOVERNORS. By AV. Trinks 

and C. Housum. Illustrated. 

No. 123. FURNACE DRAFT: ITS PRODUCTION 

by Mechanical Methods. A Handy Reference 
Book, with figures and tables. By William 
Wallace Christie. Illustrated. Second edi¬ 
tion, revised. 

























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